| L(s) = 1 | − 2.76·3-s + 4.62·9-s + 4.49·11-s − 0.103·13-s − 2·17-s − 7.25·19-s + 5.52·23-s − 4.49·27-s + 29-s − 6.76·31-s − 12.4·33-s − 5.25·37-s + 0.284·39-s + 5.79·41-s + 10.0·43-s − 11.5·47-s − 7·49-s + 5.52·51-s + 7.14·53-s + 20.0·57-s − 1.52·59-s + 9.04·61-s − 15.0·67-s − 15.2·69-s − 12.0·71-s + 1.79·73-s − 1.98·79-s + ⋯ |
| L(s) = 1 | − 1.59·3-s + 1.54·9-s + 1.35·11-s − 0.0285·13-s − 0.485·17-s − 1.66·19-s + 1.15·23-s − 0.864·27-s + 0.185·29-s − 1.21·31-s − 2.15·33-s − 0.863·37-s + 0.0455·39-s + 0.904·41-s + 1.52·43-s − 1.68·47-s − 49-s + 0.773·51-s + 0.982·53-s + 2.65·57-s − 0.198·59-s + 1.15·61-s − 1.83·67-s − 1.83·69-s − 1.42·71-s + 0.209·73-s − 0.223·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 2.76T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4.49T + 11T^{2} \) |
| 13 | \( 1 + 0.103T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 31 | \( 1 + 6.76T + 31T^{2} \) |
| 37 | \( 1 + 5.25T + 37T^{2} \) |
| 41 | \( 1 - 5.79T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 9.04T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 - 1.79T + 73T^{2} \) |
| 79 | \( 1 + 1.98T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 1.25T + 97T^{2} \) |
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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
−7.40882458862717195103705941290, −6.86742809281529718243326659571, −6.24929171740627311723306099788, −5.80103673456341173606736446237, −4.78530834962835682199833995078, −4.37416452124571840295441989906, −3.45876910125701459168782054929, −2.06205706307772085280128651028, −1.11650776132960233265609010863, 0,
1.11650776132960233265609010863, 2.06205706307772085280128651028, 3.45876910125701459168782054929, 4.37416452124571840295441989906, 4.78530834962835682199833995078, 5.80103673456341173606736446237, 6.24929171740627311723306099788, 6.86742809281529718243326659571, 7.40882458862717195103705941290
