| L(s) = 1 | − 3.21·3-s + 2.43·7-s + 7.36·9-s − 0.884·11-s − 5.10·13-s − 0.366·17-s − 2.79·19-s − 7.82·21-s − 23-s − 14.0·27-s + 8.02·29-s + 7.24·31-s + 2.84·33-s + 3.10·37-s + 16.4·39-s − 3.47·41-s − 8.56·43-s − 5.25·47-s − 1.08·49-s + 1.18·51-s + 11.6·53-s + 9.00·57-s − 9.33·59-s + 5.46·61-s + 17.9·63-s − 1.49·67-s + 3.21·69-s + ⋯ |
| L(s) = 1 | − 1.85·3-s + 0.919·7-s + 2.45·9-s − 0.266·11-s − 1.41·13-s − 0.0889·17-s − 0.642·19-s − 1.70·21-s − 0.208·23-s − 2.70·27-s + 1.49·29-s + 1.30·31-s + 0.495·33-s + 0.509·37-s + 2.63·39-s − 0.542·41-s − 1.30·43-s − 0.766·47-s − 0.155·49-s + 0.165·51-s + 1.59·53-s + 1.19·57-s − 1.21·59-s + 0.699·61-s + 2.25·63-s − 0.182·67-s + 0.387·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 3.21T + 3T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 + 0.884T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 + 0.366T + 17T^{2} \) |
| 19 | \( 1 + 2.79T + 19T^{2} \) |
| 29 | \( 1 - 8.02T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 - 3.10T + 37T^{2} \) |
| 41 | \( 1 + 3.47T + 41T^{2} \) |
| 43 | \( 1 + 8.56T + 43T^{2} \) |
| 47 | \( 1 + 5.25T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 - 8.29T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 6.06T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 - 6.55T + 97T^{2} \) |
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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
−8.448803612106727968246430757780, −7.73905958906777439805684407050, −6.80437638821351912934041905206, −6.34793555069759361341256912916, −5.20637067409355167185759853345, −4.91927384144309535324160448866, −4.19334588455954268309760167598, −2.47875295400541357402087189820, −1.26873676363799442088800438385, 0,
1.26873676363799442088800438385, 2.47875295400541357402087189820, 4.19334588455954268309760167598, 4.91927384144309535324160448866, 5.20637067409355167185759853345, 6.34793555069759361341256912916, 6.80437638821351912934041905206, 7.73905958906777439805684407050, 8.448803612106727968246430757780
