L(s) = 1 | − 2·4-s − 3·5-s + 7-s − 6·11-s + 4·16-s + 2·19-s + 6·20-s + 3·23-s + 4·25-s − 2·28-s − 29-s − 4·31-s − 3·35-s − 4·37-s − 6·41-s − 10·43-s + 12·44-s − 12·47-s + 49-s − 6·53-s + 18·55-s − 3·59-s − 4·61-s − 8·64-s − 13·67-s − 3·71-s + 2·73-s + ⋯ |
L(s) = 1 | − 4-s − 1.34·5-s + 0.377·7-s − 1.80·11-s + 16-s + 0.458·19-s + 1.34·20-s + 0.625·23-s + 4/5·25-s − 0.377·28-s − 0.185·29-s − 0.718·31-s − 0.507·35-s − 0.657·37-s − 0.937·41-s − 1.52·43-s + 1.80·44-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 2.42·55-s − 0.390·59-s − 0.512·61-s − 64-s − 1.58·67-s − 0.356·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308763 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308763 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.02189256395524, −12.39425581122305, −11.99568264207929, −11.63059117437902, −10.90604970772941, −10.73208792530829, −10.21665822416940, −9.639831600337149, −9.202138006670825, −8.577084330964265, −8.180528542615010, −7.911455658915176, −7.534251285608256, −7.004157520243834, −6.362868927428968, −5.555890164272300, −5.140705016064257, −4.854233934977166, −4.442044128299599, −3.669537511347999, −3.263050988910728, −2.979755931188376, −1.964198590821368, −1.370235369758176, −0.4126744088756177, 0,
0.4126744088756177, 1.370235369758176, 1.964198590821368, 2.979755931188376, 3.263050988910728, 3.669537511347999, 4.442044128299599, 4.854233934977166, 5.140705016064257, 5.555890164272300, 6.362868927428968, 7.004157520243834, 7.534251285608256, 7.911455658915176, 8.180528542615010, 8.577084330964265, 9.202138006670825, 9.639831600337149, 10.21665822416940, 10.73208792530829, 10.90604970772941, 11.63059117437902, 11.99568264207929, 12.39425581122305, 13.02189256395524