| L(s) = 1 | + 3-s + 7-s − 11-s + 13-s + 21-s + 23-s + 25-s − 27-s − 31-s − 33-s + 37-s + 39-s + 41-s − 47-s + 49-s − 61-s − 67-s + 69-s + 73-s + 75-s − 77-s + 79-s − 81-s − 2·89-s + 91-s − 93-s + 97-s + ⋯ |
| L(s) = 1 | + 3-s + 7-s − 11-s + 13-s + 21-s + 23-s + 25-s − 27-s − 31-s − 33-s + 37-s + 39-s + 41-s − 47-s + 49-s − 61-s − 67-s + 69-s + 73-s + 75-s − 77-s + 79-s − 81-s − 2·89-s + 91-s − 93-s + 97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.876261262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876261262\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.936292470752366358584482727446, −8.061426143527815949409199050393, −7.83558057448692456339449781803, −6.84508484969604437934462631034, −5.75856900043877506926011415177, −5.06761928239684020315908610535, −4.15254036418274628007102047858, −3.15995442538955278034255719291, −2.46859973988232204585225835820, −1.35450855192576265812662980165,
1.35450855192576265812662980165, 2.46859973988232204585225835820, 3.15995442538955278034255719291, 4.15254036418274628007102047858, 5.06761928239684020315908610535, 5.75856900043877506926011415177, 6.84508484969604437934462631034, 7.83558057448692456339449781803, 8.061426143527815949409199050393, 8.936292470752366358584482727446
