| L(s) = 1 | − 2.18·3-s + 0.879·5-s − 1.65·7-s + 1.77·9-s − 3.53·11-s + 4.94·13-s − 1.92·15-s + 2.59·17-s + 3.61·21-s − 4.82·23-s − 4.22·25-s + 2.68·27-s + 6.87·29-s − 2.04·31-s + 7.71·33-s − 1.45·35-s + 5.18·37-s − 10.8·39-s − 1.73·41-s + 7.98·43-s + 1.55·45-s + 2.92·47-s − 4.26·49-s − 5.67·51-s − 12.6·53-s − 3.10·55-s + 7.58·59-s + ⋯ |
| L(s) = 1 | − 1.26·3-s + 0.393·5-s − 0.624·7-s + 0.591·9-s − 1.06·11-s + 1.37·13-s − 0.496·15-s + 0.629·17-s + 0.787·21-s − 1.00·23-s − 0.845·25-s + 0.515·27-s + 1.27·29-s − 0.366·31-s + 1.34·33-s − 0.245·35-s + 0.852·37-s − 1.72·39-s − 0.271·41-s + 1.21·43-s + 0.232·45-s + 0.426·47-s − 0.609·49-s − 0.794·51-s − 1.74·53-s − 0.418·55-s + 0.987·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 - 0.879T + 5T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 + 3.53T + 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 + 2.04T + 31T^{2} \) |
| 37 | \( 1 - 5.18T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 - 2.92T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 16.0T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.73T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 + 5.87T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.263090766114360948452447242709, −7.66771165971377169882993394206, −6.46629723713251312982080041924, −6.08514423385117476452260731951, −5.54608864784582521472175667019, −4.65708652413809832592916483116, −3.62945736263725863025645023026, −2.61121391180836378220404653151, −1.23937457787683583822921354658, 0,
1.23937457787683583822921354658, 2.61121391180836378220404653151, 3.62945736263725863025645023026, 4.65708652413809832592916483116, 5.54608864784582521472175667019, 6.08514423385117476452260731951, 6.46629723713251312982080041924, 7.66771165971377169882993394206, 8.263090766114360948452447242709
