| L(s) = 1 | + 2·2-s − 3·3-s + 6·5-s − 6·6-s − 8·8-s + 12·10-s − 12·11-s − 76·13-s − 18·15-s − 16·16-s − 126·17-s + 20·19-s − 24·22-s − 168·23-s + 24·24-s + 125·25-s − 152·26-s + 27·27-s + 60·29-s − 36·30-s − 88·31-s + 36·33-s − 252·34-s − 254·37-s + 40·38-s + 228·39-s − 48·40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.536·5-s − 0.408·6-s − 0.353·8-s + 0.379·10-s − 0.328·11-s − 1.62·13-s − 0.309·15-s − 1/4·16-s − 1.79·17-s + 0.241·19-s − 0.232·22-s − 1.52·23-s + 0.204·24-s + 25-s − 1.14·26-s + 0.192·27-s + 0.384·29-s − 0.219·30-s − 0.509·31-s + 0.189·33-s − 1.27·34-s − 1.12·37-s + 0.170·38-s + 0.936·39-s − 0.189·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86436 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5653258753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5653258753\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 6 T - 89 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 126 T + 10963 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T - 6459 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 168 T + 16057 T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 88 T - 22047 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 254 T + 13863 T^{2} + 254 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 96 T - 94607 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 198 T - 109673 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 660 T + 230221 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 538 T + 62463 T^{2} + 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 884 T + 480693 T^{2} + 884 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 792 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 218 T - 341493 T^{2} - 218 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 520 T - 222639 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 492 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 810 T - 48869 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1154 T + p^{3} T^{2} )^{2} \) |
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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
−12.15216547835482556883230330950, −11.05662783862247437044188632211, −10.74544395123910421163138073958, −10.23664866109193176100804409578, −9.790468505158552916407266786254, −9.115861657262346682762043039465, −9.017334210720273120144212115221, −8.043162932754414211602996240800, −7.78120649299096865921239286355, −6.79539700521427949577129819070, −6.72455575328038498838498487116, −6.07233260066786283124787533782, −5.44477504019580419359726111075, −4.87102919833459958809005499664, −4.72013885712933269044801404359, −3.91458159202108903596718026235, −3.05458293498778730092072136397, −2.36501203837609088813229901009, −1.76082689393026241595696109359, −0.24399738006209510674880609782,
0.24399738006209510674880609782, 1.76082689393026241595696109359, 2.36501203837609088813229901009, 3.05458293498778730092072136397, 3.91458159202108903596718026235, 4.72013885712933269044801404359, 4.87102919833459958809005499664, 5.44477504019580419359726111075, 6.07233260066786283124787533782, 6.72455575328038498838498487116, 6.79539700521427949577129819070, 7.78120649299096865921239286355, 8.043162932754414211602996240800, 9.017334210720273120144212115221, 9.115861657262346682762043039465, 9.790468505158552916407266786254, 10.23664866109193176100804409578, 10.74544395123910421163138073958, 11.05662783862247437044188632211, 12.15216547835482556883230330950
