| L(s) = 1 | − 1.41·5-s − 3·9-s + 7.07·13-s − 8·17-s − 2.99·25-s − 4.24·29-s − 9.89·37-s − 8·41-s + 4.24·45-s − 7·49-s − 12.7·53-s + 15.5·61-s − 10.0·65-s + 6·73-s + 9·81-s + 11.3·85-s − 10·89-s − 8·97-s + 12.7·101-s − 18.3·109-s + 14·113-s − 21.2·117-s + ⋯ |
| L(s) = 1 | − 0.632·5-s − 9-s + 1.96·13-s − 1.94·17-s − 0.599·25-s − 0.787·29-s − 1.62·37-s − 1.24·41-s + 0.632·45-s − 49-s − 1.74·53-s + 1.99·61-s − 1.24·65-s + 0.702·73-s + 81-s + 1.22·85-s − 1.05·89-s − 0.812·97-s + 1.26·101-s − 1.76·109-s + 1.31·113-s − 1.96·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7.07T + 13T^{2} \) |
| 17 | \( 1 + 8T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.317675711835159939530734911221, −8.541313828150603784925593638848, −8.187858025839315248535384461962, −6.86394040640783585532286938758, −6.21163643331104193345533078788, −5.20724840353972805445065672722, −4.01968561887729108274197437874, −3.30830255368212510429608639584, −1.86135918480934937808949053790, 0,
1.86135918480934937808949053790, 3.30830255368212510429608639584, 4.01968561887729108274197437874, 5.20724840353972805445065672722, 6.21163643331104193345533078788, 6.86394040640783585532286938758, 8.187858025839315248535384461962, 8.541313828150603784925593638848, 9.317675711835159939530734911221
