| L(s) = 1 | − 1.98·2-s − 3-s + 1.94·4-s − 5-s + 1.98·6-s − 2.98·7-s + 0.111·8-s + 9-s + 1.98·10-s + 5.57·11-s − 1.94·12-s − 2.87·13-s + 5.92·14-s + 15-s − 4.10·16-s + 6.23·17-s − 1.98·18-s − 3.43·19-s − 1.94·20-s + 2.98·21-s − 11.0·22-s − 5.27·23-s − 0.111·24-s + 25-s + 5.71·26-s − 27-s − 5.79·28-s + ⋯ |
| L(s) = 1 | − 1.40·2-s − 0.577·3-s + 0.972·4-s − 0.447·5-s + 0.810·6-s − 1.12·7-s + 0.0392·8-s + 0.333·9-s + 0.628·10-s + 1.68·11-s − 0.561·12-s − 0.798·13-s + 1.58·14-s + 0.258·15-s − 1.02·16-s + 1.51·17-s − 0.468·18-s − 0.787·19-s − 0.434·20-s + 0.650·21-s − 2.35·22-s − 1.09·23-s − 0.0226·24-s + 0.200·25-s + 1.12·26-s − 0.192·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.98T + 2T^{2} \) |
| 7 | \( 1 + 2.98T + 7T^{2} \) |
| 11 | \( 1 - 5.57T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 19 | \( 1 + 3.43T + 19T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 0.153T + 31T^{2} \) |
| 37 | \( 1 - 2.03T + 37T^{2} \) |
| 41 | \( 1 + 7.80T + 41T^{2} \) |
| 43 | \( 1 - 6.57T + 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 + 7.15T + 53T^{2} \) |
| 59 | \( 1 + 4.60T + 59T^{2} \) |
| 61 | \( 1 - 0.941T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 0.369T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 + 3.19T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.82732883320398540986980393381, −7.06664687876323690729415969064, −6.52039051563111210203269885329, −5.97336422772213474114965677076, −4.76816658479699378514681068571, −3.99376701786905117988765315027, −3.19148525889629296326757626661, −1.90620808163720058819136934824, −0.943914904018508483610545414665, 0,
0.943914904018508483610545414665, 1.90620808163720058819136934824, 3.19148525889629296326757626661, 3.99376701786905117988765315027, 4.76816658479699378514681068571, 5.97336422772213474114965677076, 6.52039051563111210203269885329, 7.06664687876323690729415969064, 7.82732883320398540986980393381
