| L(s) = 1 | + 1.43·2-s − 5.94·4-s + 5·5-s − 23.1·7-s − 19.9·8-s + 7.17·10-s + 11·11-s − 85.3·13-s − 33.1·14-s + 18.8·16-s + 109.·17-s + 121.·19-s − 29.7·20-s + 15.7·22-s + 171.·23-s + 25·25-s − 122.·26-s + 137.·28-s + 80.1·29-s − 255.·31-s + 187.·32-s + 156.·34-s − 115.·35-s + 103.·37-s + 174.·38-s − 99.9·40-s − 194.·41-s + ⋯ |
| L(s) = 1 | + 0.507·2-s − 0.742·4-s + 0.447·5-s − 1.24·7-s − 0.883·8-s + 0.226·10-s + 0.301·11-s − 1.82·13-s − 0.632·14-s + 0.294·16-s + 1.55·17-s + 1.46·19-s − 0.332·20-s + 0.152·22-s + 1.55·23-s + 0.200·25-s − 0.923·26-s + 0.926·28-s + 0.513·29-s − 1.47·31-s + 1.03·32-s + 0.789·34-s − 0.557·35-s + 0.460·37-s + 0.743·38-s − 0.395·40-s − 0.741·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.677517850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677517850\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 1.43T + 8T^{2} \) |
| 7 | \( 1 + 23.1T + 343T^{2} \) |
| 13 | \( 1 + 85.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 171.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 80.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 255.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 434.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 459.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 656.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 158.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 107.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 685.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 671.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 260.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 361.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 706.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 557.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 482.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.16902861205828707539026681730, −9.597841876156844087900740509012, −9.157330393058536025024612876081, −7.65873681140022787171825107164, −6.79808802252375812300487166897, −5.54605089910991316104142476394, −5.04499595334963064646950193070, −3.56403289830753289965216662070, −2.82863176035414382923284159853, −0.74617468037257595814509702962,
0.74617468037257595814509702962, 2.82863176035414382923284159853, 3.56403289830753289965216662070, 5.04499595334963064646950193070, 5.54605089910991316104142476394, 6.79808802252375812300487166897, 7.65873681140022787171825107164, 9.157330393058536025024612876081, 9.597841876156844087900740509012, 10.16902861205828707539026681730
