L(s) = 1 | + 4-s − 2·5-s + 9-s − 3·11-s − 3·16-s − 2·20-s − 25-s + 4·29-s + 36-s − 4·41-s − 3·44-s − 2·45-s + 2·49-s + 6·55-s + 24·59-s − 4·61-s − 7·64-s + 24·79-s + 6·80-s + 81-s − 12·89-s − 3·99-s − 100-s − 12·101-s − 20·109-s + 4·116-s + 2·121-s + ⋯ |
L(s) = 1 | + 1/2·4-s − 0.894·5-s + 1/3·9-s − 0.904·11-s − 3/4·16-s − 0.447·20-s − 1/5·25-s + 0.742·29-s + 1/6·36-s − 0.624·41-s − 0.452·44-s − 0.298·45-s + 2/7·49-s + 0.809·55-s + 3.12·59-s − 0.512·61-s − 7/8·64-s + 2.70·79-s + 0.670·80-s + 1/9·81-s − 1.27·89-s − 0.301·99-s − 0.0999·100-s − 1.19·101-s − 1.91·109-s + 0.371·116-s + 2/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6606441068\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6606441068\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
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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
−13.14575241523544191772257609292, −12.23662699462035013540508107849, −11.93851842213943004021314056240, −11.22694089429559050080949866662, −10.73683235986961041263444060107, −10.08219127334790378188359306081, −9.355764873253690620092615692808, −8.396627232303075215483887640772, −7.987064380804719908863906390256, −7.14294632569963089923076678235, −6.67514988045421682235703457049, −5.55778831301744703808824925181, −4.66431952886322862521445388232, −3.71584900926351816916902784454, −2.45765074847302232037845071651,
2.45765074847302232037845071651, 3.71584900926351816916902784454, 4.66431952886322862521445388232, 5.55778831301744703808824925181, 6.67514988045421682235703457049, 7.14294632569963089923076678235, 7.987064380804719908863906390256, 8.396627232303075215483887640772, 9.355764873253690620092615692808, 10.08219127334790378188359306081, 10.73683235986961041263444060107, 11.22694089429559050080949866662, 11.93851842213943004021314056240, 12.23662699462035013540508107849, 13.14575241523544191772257609292