| L(s) = 1 | − 4-s − 9-s + 16-s − 4·19-s − 8·31-s + 36-s − 12·41-s + 10·49-s − 20·61-s − 64-s + 4·76-s − 16·79-s + 81-s − 12·89-s − 24·101-s + 32·109-s − 22·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1/4·16-s − 0.917·19-s − 1.43·31-s + 1/6·36-s − 1.87·41-s + 10/7·49-s − 2.56·61-s − 1/8·64-s + 0.458·76-s − 1.80·79-s + 1/9·81-s − 1.27·89-s − 2.38·101-s + 3.06·109-s − 2·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6355796214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6355796214\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.230611402726732812170573677288, −8.951924251777135754726001946533, −8.657938088058476386904902920214, −8.346502302177859191431134507207, −7.84996991225705324931273227250, −7.32696566167836058548444849898, −7.22842422311618937493395305917, −6.46494201568305731324293918266, −6.32073832279531656660949266772, −5.68684127396678249394759724577, −5.40890418722005765692201476575, −4.94913320452898524656789408510, −4.49121222152920998992895410726, −3.91828874944976649504861745303, −3.75096483742186052697352024645, −2.94582154192333387566604590077, −2.68121323463964597993330153684, −1.79554827177164743504482470586, −1.45010747657485842520841235924, −0.28860504798558845416016847759,
0.28860504798558845416016847759, 1.45010747657485842520841235924, 1.79554827177164743504482470586, 2.68121323463964597993330153684, 2.94582154192333387566604590077, 3.75096483742186052697352024645, 3.91828874944976649504861745303, 4.49121222152920998992895410726, 4.94913320452898524656789408510, 5.40890418722005765692201476575, 5.68684127396678249394759724577, 6.32073832279531656660949266772, 6.46494201568305731324293918266, 7.22842422311618937493395305917, 7.32696566167836058548444849898, 7.84996991225705324931273227250, 8.346502302177859191431134507207, 8.657938088058476386904902920214, 8.951924251777135754726001946533, 9.230611402726732812170573677288
