| L(s) = 1 | + 2.56·3-s − 2.64·5-s + 0.180·7-s + 3.56·9-s − 0.644·11-s + 3.94·13-s − 6.77·15-s + 5.56·17-s + 19-s + 0.463·21-s + 4.46·23-s + 1.99·25-s + 1.43·27-s + 3.94·29-s + 5.48·31-s − 1.65·33-s − 0.478·35-s − 7.48·37-s + 10.1·39-s + 8.41·41-s − 5.76·43-s − 9.41·45-s + 11.2·47-s − 6.96·49-s + 14.2·51-s − 7.58·53-s + 1.70·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 1.18·5-s + 0.0683·7-s + 1.18·9-s − 0.194·11-s + 1.09·13-s − 1.74·15-s + 1.35·17-s + 0.229·19-s + 0.101·21-s + 0.930·23-s + 0.398·25-s + 0.276·27-s + 0.733·29-s + 0.985·31-s − 0.287·33-s − 0.0808·35-s − 1.23·37-s + 1.61·39-s + 1.31·41-s − 0.879·43-s − 1.40·45-s + 1.64·47-s − 0.995·49-s + 1.99·51-s − 1.04·53-s + 0.229·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.434226596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434226596\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 7 | \( 1 - 0.180T + 7T^{2} \) |
| 11 | \( 1 + 0.644T + 11T^{2} \) |
| 13 | \( 1 - 3.94T + 13T^{2} \) |
| 17 | \( 1 - 5.56T + 17T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 8.41T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 2.47T + 61T^{2} \) |
| 67 | \( 1 + 8.97T + 67T^{2} \) |
| 71 | \( 1 - 7.63T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 17.1T + 89T^{2} \) |
| 97 | \( 1 - 3.28T + 97T^{2} \) |
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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
−9.542707087762264654244879569118, −8.673286864719519094449754319950, −8.143818285277217066755242438123, −7.63051404706707681983645115393, −6.70881497113278522300333750110, −5.37799706097445190396062840311, −4.16394025055850417656785202711, −3.47557769501295184170228693616, −2.77990280888157109607639758188, −1.19533803010128349088953861527,
1.19533803010128349088953861527, 2.77990280888157109607639758188, 3.47557769501295184170228693616, 4.16394025055850417656785202711, 5.37799706097445190396062840311, 6.70881497113278522300333750110, 7.63051404706707681983645115393, 8.143818285277217066755242438123, 8.673286864719519094449754319950, 9.542707087762264654244879569118
