| L(s) = 1 | + 512·2-s + 1.96e5·4-s − 5.02e5·5-s − 8.89e6·7-s + 6.71e7·8-s − 2.57e8·10-s + 9.63e8·11-s + 3.54e9·13-s − 4.55e9·14-s + 2.14e10·16-s − 7.60e9·17-s + 1.34e11·19-s − 9.88e10·20-s + 4.93e11·22-s + 4.59e11·23-s + 6.81e10·25-s + 1.81e12·26-s − 1.74e12·28-s + 1.27e12·29-s − 7.79e12·31-s + 6.59e12·32-s − 3.89e12·34-s + 4.47e12·35-s − 2.34e13·37-s + 6.91e13·38-s − 3.37e13·40-s + 1.68e14·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.575·5-s − 0.583·7-s + 1.41·8-s − 0.813·10-s + 1.35·11-s + 1.20·13-s − 0.824·14-s + 5/4·16-s − 0.264·17-s + 1.82·19-s − 0.863·20-s + 1.91·22-s + 1.22·23-s + 0.0893·25-s + 1.70·26-s − 0.874·28-s + 0.474·29-s − 1.64·31-s + 1.06·32-s − 0.374·34-s + 0.335·35-s − 1.09·37-s + 2.57·38-s − 0.813·40-s + 3.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+17/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(9.197261875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.197261875\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{8} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 502656 T + 7380378394 p^{2} T^{2} + 502656 p^{17} T^{3} + p^{34} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 8894600 T + 2560785540990 p^{2} T^{2} + 8894600 p^{17} T^{3} + p^{34} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 7964160 p^{2} T + 9172033030498246 p^{2} T^{2} - 7964160 p^{19} T^{3} + p^{34} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3542987140 T + 753982175104565910 p T^{2} - 3542987140 p^{17} T^{3} + p^{34} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7609529088 T + \)\(83\!\cdots\!90\)\( T^{2} + 7609529088 p^{17} T^{3} + p^{34} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 134964738832 T + \)\(11\!\cdots\!34\)\( T^{2} - 134964738832 p^{17} T^{3} + p^{34} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 459827512320 T + \)\(33\!\cdots\!06\)\( T^{2} - 459827512320 p^{17} T^{3} + p^{34} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 1278177993600 T + \)\(10\!\cdots\!62\)\( T^{2} - 1278177993600 p^{17} T^{3} + p^{34} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7792392120968 T + \)\(51\!\cdots\!78\)\( T^{2} + 7792392120968 p^{17} T^{3} + p^{34} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 23466992522900 T + \)\(94\!\cdots\!10\)\( T^{2} + 23466992522900 p^{17} T^{3} + p^{34} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 168365456474880 T + \)\(12\!\cdots\!62\)\( T^{2} - 168365456474880 p^{17} T^{3} + p^{34} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 49921265397520 T + \)\(11\!\cdots\!02\)\( T^{2} - 49921265397520 p^{17} T^{3} + p^{34} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 391426169164800 T + \)\(90\!\cdots\!74\)\( T^{2} - 391426169164800 p^{17} T^{3} + p^{34} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 634140086481792 T + \)\(40\!\cdots\!42\)\( T^{2} - 634140086481792 p^{17} T^{3} + p^{34} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 704352386411520 T - \)\(33\!\cdots\!78\)\( T^{2} + 704352386411520 p^{17} T^{3} + p^{34} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2623204591588196 T + \)\(43\!\cdots\!46\)\( T^{2} + 2623204591588196 p^{17} T^{3} + p^{34} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3189217530029600 T + \)\(24\!\cdots\!58\)\( T^{2} + 3189217530029600 p^{17} T^{3} + p^{34} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6461300085166080 T + \)\(59\!\cdots\!78\)\( T^{2} + 6461300085166080 p^{17} T^{3} + p^{34} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5119332487322900 T + \)\(11\!\cdots\!06\)\( T^{2} + 5119332487322900 p^{17} T^{3} + p^{34} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3448490746896008 T - \)\(24\!\cdots\!66\)\( T^{2} + 3448490746896008 p^{17} T^{3} + p^{34} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6068774704022016 T + \)\(43\!\cdots\!10\)\( T^{2} - 6068774704022016 p^{17} T^{3} + p^{34} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 116927717083752960 T + \)\(60\!\cdots\!14\)\( T^{2} - 116927717083752960 p^{17} T^{3} + p^{34} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 52820276144283620 T + \)\(10\!\cdots\!70\)\( T^{2} + 52820276144283620 p^{17} T^{3} + p^{34} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.71426432290310047683524166908, −14.29583918677957384698163594445, −13.57610664977680789261113552878, −13.11802148789651094021908414313, −12.08110436874153781909441236063, −12.05393014610344216466029094132, −10.97742461784294834261578275446, −10.73948660388654417432380910549, −9.205195209392530108210187442832, −9.010241132237638757630278567520, −7.33375641157934520527144797689, −7.31927610304612673525897854765, −6.09310732120741865016802057312, −5.75988569141101405478205300666, −4.62463859840070316167652162270, −3.86210494666753675812822243190, −3.44063985502420303296377966946, −2.63627726635458109618891076207, −1.34696200294309980975653610899, −0.824503678306524575608183618627,
0.824503678306524575608183618627, 1.34696200294309980975653610899, 2.63627726635458109618891076207, 3.44063985502420303296377966946, 3.86210494666753675812822243190, 4.62463859840070316167652162270, 5.75988569141101405478205300666, 6.09310732120741865016802057312, 7.31927610304612673525897854765, 7.33375641157934520527144797689, 9.010241132237638757630278567520, 9.205195209392530108210187442832, 10.73948660388654417432380910549, 10.97742461784294834261578275446, 12.05393014610344216466029094132, 12.08110436874153781909441236063, 13.11802148789651094021908414313, 13.57610664977680789261113552878, 14.29583918677957384698163594445, 14.71426432290310047683524166908
