| L(s) = 1 | − 4·4-s − 10·5-s + 50·9-s − 56·11-s + 16·16-s + 120·19-s + 40·20-s − 25·25-s − 180·29-s − 256·31-s − 200·36-s + 484·41-s + 224·44-s − 500·45-s + 10·49-s + 560·55-s + 40·59-s + 1.08e3·61-s − 64·64-s − 2.25e3·71-s − 480·76-s + 1.44e3·79-s − 160·80-s + 1.77e3·81-s + 980·89-s − 1.20e3·95-s − 2.80e3·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s + 1.85·9-s − 1.53·11-s + 1/4·16-s + 1.44·19-s + 0.447·20-s − 1/5·25-s − 1.15·29-s − 1.48·31-s − 0.925·36-s + 1.84·41-s + 0.767·44-s − 1.65·45-s + 0.0291·49-s + 1.37·55-s + 0.0882·59-s + 2.27·61-s − 1/8·64-s − 3.77·71-s − 0.724·76-s + 2.05·79-s − 0.223·80-s + 2.42·81-s + 1.16·89-s − 1.29·95-s − 2.84·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6642845131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6642845131\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 5730 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 60 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 128 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 45610 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 242 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 27970 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 156570 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 286090 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 542 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 413170 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1128 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 378610 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 915090 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 490 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 294590 T^{2} + p^{6} T^{4} \) |
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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
−20.75404921372514853975775707566, −20.49393296500168893853370391538, −19.24234170159821737585396352626, −18.93994862899296652043141094959, −18.00134541967902288543044622023, −17.98633281071982953329448072587, −16.21971445973795486680043857101, −16.12713599752384189961533448556, −15.36501571992824630772995142126, −14.56566335222668874528233179028, −13.24272907549937557694067291051, −13.04587947367273779802550776539, −12.08341892017131722999543459693, −10.98819281810985439264008946563, −10.10423520548203916488377750395, −9.264129044356701969990396274366, −7.68917674492804863897978674182, −7.43215939908228079028033812646, −5.30666260524625395791180327754, −3.96135615060000715278264354271,
3.96135615060000715278264354271, 5.30666260524625395791180327754, 7.43215939908228079028033812646, 7.68917674492804863897978674182, 9.264129044356701969990396274366, 10.10423520548203916488377750395, 10.98819281810985439264008946563, 12.08341892017131722999543459693, 13.04587947367273779802550776539, 13.24272907549937557694067291051, 14.56566335222668874528233179028, 15.36501571992824630772995142126, 16.12713599752384189961533448556, 16.21971445973795486680043857101, 17.98633281071982953329448072587, 18.00134541967902288543044622023, 18.93994862899296652043141094959, 19.24234170159821737585396352626, 20.49393296500168893853370391538, 20.75404921372514853975775707566
