| L(s) = 1 | − 3.30·3-s − 0.852·7-s + 7.94·9-s − 1.80·11-s − 2.70·13-s − 4.05·17-s − 7.19·19-s + 2.82·21-s + 23-s − 16.3·27-s − 3.76·29-s − 4.00·31-s + 5.98·33-s − 11.4·37-s + 8.95·39-s + 6.09·41-s − 11.4·43-s + 7.31·47-s − 6.27·49-s + 13.4·51-s − 0.406·53-s + 23.8·57-s + 4.83·59-s + 9.22·61-s − 6.77·63-s − 9.64·67-s − 3.30·69-s + ⋯ |
| L(s) = 1 | − 1.91·3-s − 0.322·7-s + 2.64·9-s − 0.545·11-s − 0.750·13-s − 0.982·17-s − 1.65·19-s + 0.615·21-s + 0.208·23-s − 3.15·27-s − 0.698·29-s − 0.719·31-s + 1.04·33-s − 1.88·37-s + 1.43·39-s + 0.951·41-s − 1.73·43-s + 1.06·47-s − 0.896·49-s + 1.87·51-s − 0.0558·53-s + 3.15·57-s + 0.629·59-s + 1.18·61-s − 0.854·63-s − 1.17·67-s − 0.398·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03990354646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03990354646\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.30T + 3T^{2} \) |
| 7 | \( 1 + 0.852T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 2.70T + 13T^{2} \) |
| 17 | \( 1 + 4.05T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 + 4.00T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 6.09T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 7.31T + 47T^{2} \) |
| 53 | \( 1 + 0.406T + 53T^{2} \) |
| 59 | \( 1 - 4.83T + 59T^{2} \) |
| 61 | \( 1 - 9.22T + 61T^{2} \) |
| 67 | \( 1 + 9.64T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 + 8.60T + 79T^{2} \) |
| 83 | \( 1 - 2.49T + 83T^{2} \) |
| 89 | \( 1 + 5.41T + 89T^{2} \) |
| 97 | \( 1 + 8.41T + 97T^{2} \) |
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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
−7.34720674759960617685939656665, −6.85909723883176714328784377290, −6.39960728126231903684395965648, −5.58090586702448885614523704255, −5.12227737837710568576549730680, −4.42829292620684011417454258446, −3.77198157617030459985358838589, −2.39642756770163140212982733361, −1.58197238036161818795958207185, −0.10455732064116207282758968870,
0.10455732064116207282758968870, 1.58197238036161818795958207185, 2.39642756770163140212982733361, 3.77198157617030459985358838589, 4.42829292620684011417454258446, 5.12227737837710568576549730680, 5.58090586702448885614523704255, 6.39960728126231903684395965648, 6.85909723883176714328784377290, 7.34720674759960617685939656665
