| L(s) = 1 | + 4·5-s + 5·9-s − 2·11-s − 2·19-s + 11·25-s + 2·29-s − 2·31-s + 20·45-s + 13·49-s − 8·55-s − 8·59-s − 14·61-s + 10·71-s − 8·79-s + 16·81-s + 14·89-s − 8·95-s − 10·99-s − 20·101-s − 20·109-s + 3·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 5/3·9-s − 0.603·11-s − 0.458·19-s + 11/5·25-s + 0.371·29-s − 0.359·31-s + 2.98·45-s + 13/7·49-s − 1.07·55-s − 1.04·59-s − 1.79·61-s + 1.18·71-s − 0.900·79-s + 16/9·81-s + 1.48·89-s − 0.820·95-s − 1.00·99-s − 1.99·101-s − 1.91·109-s + 3/11·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.694817584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.694817584\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96204283325738727631663142423, −10.82363709223638425886916429788, −10.36437055045198716504439160744, −10.05228025991007960734073145114, −9.596895731312918570686994084414, −9.239515366937217046208034029278, −8.822716182437411563588507555940, −8.173451757540625524449103432871, −7.54555479629257307603786610140, −7.20401396368338010417614047920, −6.51874874837716799735560030318, −6.34779806164796797682815056701, −5.56528834141212959111599732244, −5.27540683747586444306474300080, −4.54527580989995364796753291808, −4.17671272069787922677323321393, −3.21614344233243892520096554711, −2.46255277453660539042908429911, −1.88186128453870152472191643390, −1.19025606975213521752961924701,
1.19025606975213521752961924701, 1.88186128453870152472191643390, 2.46255277453660539042908429911, 3.21614344233243892520096554711, 4.17671272069787922677323321393, 4.54527580989995364796753291808, 5.27540683747586444306474300080, 5.56528834141212959111599732244, 6.34779806164796797682815056701, 6.51874874837716799735560030318, 7.20401396368338010417614047920, 7.54555479629257307603786610140, 8.173451757540625524449103432871, 8.822716182437411563588507555940, 9.239515366937217046208034029278, 9.596895731312918570686994084414, 10.05228025991007960734073145114, 10.36437055045198716504439160744, 10.82363709223638425886916429788, 10.96204283325738727631663142423
