| L(s) = 1 | − 4·4-s + 50·9-s − 40·11-s + 16·16-s + 38·19-s + 300·29-s − 400·31-s − 200·36-s − 436·41-s + 160·44-s + 542·49-s + 96·59-s − 268·61-s − 64·64-s − 1.04e3·71-s − 152·76-s − 1.96e3·79-s + 1.77e3·81-s − 1.34e3·89-s − 2.00e3·99-s − 2.90e3·101-s − 204·109-s − 1.20e3·116-s − 1.46e3·121-s + 1.60e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.85·9-s − 1.09·11-s + 1/4·16-s + 0.458·19-s + 1.92·29-s − 2.31·31-s − 0.925·36-s − 1.66·41-s + 0.548·44-s + 1.58·49-s + 0.211·59-s − 0.562·61-s − 1/8·64-s − 1.73·71-s − 0.229·76-s − 2.79·79-s + 2.42·81-s − 1.59·89-s − 2.03·99-s − 2.86·101-s − 0.179·109-s − 0.960·116-s − 1.09·121-s + 1.15·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.242543463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242543463\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 542 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4378 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 p T + p^{3} T^{2} )( 1 + 8 p T + p^{3} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 22734 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 200 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 76970 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 218 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 97510 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 175246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 292570 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 48 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 134 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 489970 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 520 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 p T + p^{3} T^{2} )( 1 + 16 p T + p^{3} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 980 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1119238 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 670 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 561970 T^{2} + p^{6} T^{4} \) |
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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
−10.00680119397099416951034887447, −9.335412605397897888079242040363, −9.253971588312011306863996065350, −8.423319804463192113600298152818, −8.385449894472639760674298865620, −7.61272288859277580288664220435, −7.42143650074165131416463862982, −6.85692955742505895498820160955, −6.75666968290261896456473264749, −5.78025519074571187509642351892, −5.53367847717404774387777941741, −5.00524519247873051202993665167, −4.56000776921054146312112472072, −4.11004285251429375800000647808, −3.69345342151484724144080848080, −2.95195305480885345839012777033, −2.49479980694000286262715160334, −1.45863141572526765602674696460, −1.39225666168624050801992227113, −0.29357744761952592217075139028,
0.29357744761952592217075139028, 1.39225666168624050801992227113, 1.45863141572526765602674696460, 2.49479980694000286262715160334, 2.95195305480885345839012777033, 3.69345342151484724144080848080, 4.11004285251429375800000647808, 4.56000776921054146312112472072, 5.00524519247873051202993665167, 5.53367847717404774387777941741, 5.78025519074571187509642351892, 6.75666968290261896456473264749, 6.85692955742505895498820160955, 7.42143650074165131416463862982, 7.61272288859277580288664220435, 8.385449894472639760674298865620, 8.423319804463192113600298152818, 9.253971588312011306863996065350, 9.335412605397897888079242040363, 10.00680119397099416951034887447
