| L(s) = 1 | − 0.873·2-s − 3-s − 1.23·4-s − 5-s + 0.873·6-s + 0.198·7-s + 2.82·8-s + 9-s + 0.873·10-s + 2.72·11-s + 1.23·12-s + 13-s − 0.173·14-s + 15-s + 0.00302·16-s − 2.87·17-s − 0.873·18-s − 5.61·19-s + 1.23·20-s − 0.198·21-s − 2.38·22-s + 1.63·23-s − 2.82·24-s + 25-s − 0.873·26-s − 27-s − 0.245·28-s + ⋯ |
| L(s) = 1 | − 0.617·2-s − 0.577·3-s − 0.618·4-s − 0.447·5-s + 0.356·6-s + 0.0750·7-s + 0.999·8-s + 0.333·9-s + 0.276·10-s + 0.822·11-s + 0.357·12-s + 0.277·13-s − 0.0463·14-s + 0.258·15-s + 0.000757·16-s − 0.696·17-s − 0.205·18-s − 1.28·19-s + 0.276·20-s − 0.0433·21-s − 0.508·22-s + 0.341·23-s − 0.577·24-s + 0.200·25-s − 0.171·26-s − 0.192·27-s − 0.0464·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.873T + 2T^{2} \) |
| 7 | \( 1 - 0.198T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 17 | \( 1 + 2.87T + 17T^{2} \) |
| 19 | \( 1 + 5.61T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 37 | \( 1 + 3.28T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 - 1.06T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 1.80T + 59T^{2} \) |
| 61 | \( 1 - 7.03T + 61T^{2} \) |
| 67 | \( 1 + 1.54T + 67T^{2} \) |
| 71 | \( 1 - 5.33T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 + 7.18T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.963816876930923577297518323718, −6.86257344315280286408857330259, −6.58421164484236096510003109129, −5.55078387118744997444337915846, −4.63928686716980474877669435512, −4.26669213850469105702774046287, −3.40262010132737098844344132333, −2.00477565485617207685548362105, −1.02677608203285845917866282472, 0,
1.02677608203285845917866282472, 2.00477565485617207685548362105, 3.40262010132737098844344132333, 4.26669213850469105702774046287, 4.63928686716980474877669435512, 5.55078387118744997444337915846, 6.58421164484236096510003109129, 6.86257344315280286408857330259, 7.963816876930923577297518323718
