| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 13-s + 15-s − 17-s − 6·19-s − 2·21-s + 25-s − 27-s − 6·29-s − 4·31-s − 2·35-s − 2·37-s − 39-s − 10·41-s − 2·43-s − 45-s − 3·49-s + 51-s − 14·53-s + 6·57-s − 6·59-s − 10·61-s + 2·63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 1.37·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.160·39-s − 1.56·41-s − 0.304·43-s − 0.149·45-s − 3/7·49-s + 0.140·51-s − 1.92·53-s + 0.794·57-s − 0.781·59-s − 1.28·61-s + 0.251·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.30070968177192, −13.01481848535974, −12.43470815322462, −12.10865055787662, −11.48296983689232, −11.14096344000361, −10.79640664222695, −10.43437544598491, −9.703572460036373, −9.249417221209253, −8.702231062392874, −8.143181269249686, −7.941678971402497, −7.150712573109657, −6.870707621570298, −6.172405176067659, −5.865382363695187, −5.064593368263802, −4.776898430204708, −4.282973179637623, −3.610900692144056, −3.230090259466500, −2.227783472474129, −1.752529681293272, −1.255977236038738, 0, 0,
1.255977236038738, 1.752529681293272, 2.227783472474129, 3.230090259466500, 3.610900692144056, 4.282973179637623, 4.776898430204708, 5.064593368263802, 5.865382363695187, 6.172405176067659, 6.870707621570298, 7.150712573109657, 7.941678971402497, 8.143181269249686, 8.702231062392874, 9.249417221209253, 9.703572460036373, 10.43437544598491, 10.79640664222695, 11.14096344000361, 11.48296983689232, 12.10865055787662, 12.43470815322462, 13.01481848535974, 13.30070968177192
