| L(s) = 1 | − 36·4-s + 290·9-s + 464·11-s + 272·16-s + 196·19-s − 6.83e3·29-s − 1.52e4·31-s − 1.04e4·36-s − 3.59e4·41-s − 1.67e4·44-s − 2.40e3·49-s + 5.46e3·59-s + 5.12e4·61-s + 2.70e4·64-s − 1.17e5·71-s − 7.05e3·76-s − 6.35e4·79-s + 2.50e4·81-s + 1.01e5·89-s + 1.34e5·99-s + 7.73e4·101-s − 1.85e5·109-s + 2.46e5·116-s − 1.60e5·121-s + 5.50e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 9/8·4-s + 1.19·9-s + 1.15·11-s + 0.265·16-s + 0.124·19-s − 1.50·29-s − 2.85·31-s − 1.34·36-s − 3.33·41-s − 1.30·44-s − 1/7·49-s + 0.204·59-s + 1.76·61-s + 0.826·64-s − 2.75·71-s − 0.140·76-s − 1.14·79-s + 0.424·81-s + 1.35·89-s + 1.37·99-s + 0.754·101-s − 1.49·109-s + 1.69·116-s − 0.997·121-s + 3.21·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9623829467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9623829467\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 9 p^{2} T^{2} + p^{10} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 290 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 232 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 722986 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 125570 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 98 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9545710 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3418 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7644 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 30569510 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 17962 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 175642486 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 371753038 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 831274342 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2730 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 25648 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 357302998 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 58560 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 489015538 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 31784 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7456271842 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 50582 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 13751728478 T^{2} + p^{10} T^{4} \) |
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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
−12.91269464340591816008602117613, −11.68674291682769855490633286579, −11.12417186135891365996708532963, −10.40381455646279359682469957744, −10.00352208368061456795185530937, −9.497692167891142192140958305570, −8.960262953091358381471642166789, −8.829539497390668569175572792110, −8.013606675557466591342558635490, −7.16836310586309377145304452422, −7.10463520403413235450155151994, −6.31026300298808922525380989898, −5.38038814576310393891059234361, −5.13264298361291426951158643052, −4.17507049099793042123017350182, −3.90815879922517592590090250701, −3.32512056566408719348114438354, −1.81701390460019766170532608122, −1.52607186214411363042823684444, −0.32602610511261903622463805952,
0.32602610511261903622463805952, 1.52607186214411363042823684444, 1.81701390460019766170532608122, 3.32512056566408719348114438354, 3.90815879922517592590090250701, 4.17507049099793042123017350182, 5.13264298361291426951158643052, 5.38038814576310393891059234361, 6.31026300298808922525380989898, 7.10463520403413235450155151994, 7.16836310586309377145304452422, 8.013606675557466591342558635490, 8.829539497390668569175572792110, 8.960262953091358381471642166789, 9.497692167891142192140958305570, 10.00352208368061456795185530937, 10.40381455646279359682469957744, 11.12417186135891365996708532963, 11.68674291682769855490633286579, 12.91269464340591816008602117613
