| L(s) = 1 | + 1.10·2-s + 3-s − 0.779·4-s + 1.78·5-s + 1.10·6-s + 0.541·7-s − 3.07·8-s + 9-s + 1.96·10-s − 1.79·11-s − 0.779·12-s − 2.27·13-s + 0.597·14-s + 1.78·15-s − 1.83·16-s + 17-s + 1.10·18-s − 6.41·19-s − 1.39·20-s + 0.541·21-s − 1.98·22-s − 3.10·23-s − 3.07·24-s − 1.82·25-s − 2.50·26-s + 27-s − 0.422·28-s + ⋯ |
| L(s) = 1 | + 0.781·2-s + 0.577·3-s − 0.389·4-s + 0.797·5-s + 0.450·6-s + 0.204·7-s − 1.08·8-s + 0.333·9-s + 0.622·10-s − 0.542·11-s − 0.225·12-s − 0.630·13-s + 0.159·14-s + 0.460·15-s − 0.457·16-s + 0.242·17-s + 0.260·18-s − 1.47·19-s − 0.310·20-s + 0.118·21-s − 0.423·22-s − 0.646·23-s − 0.626·24-s − 0.364·25-s − 0.492·26-s + 0.192·27-s − 0.0798·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 0.541T + 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 - 7.57T + 29T^{2} \) |
| 31 | \( 1 - 8.37T + 31T^{2} \) |
| 37 | \( 1 - 2.19T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 + 5.40T + 43T^{2} \) |
| 47 | \( 1 + 4.05T + 47T^{2} \) |
| 53 | \( 1 + 5.17T + 53T^{2} \) |
| 59 | \( 1 + 3.24T + 59T^{2} \) |
| 61 | \( 1 + 6.06T + 61T^{2} \) |
| 67 | \( 1 + 0.291T + 67T^{2} \) |
| 71 | \( 1 + 6.46T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 4.11T + 83T^{2} \) |
| 89 | \( 1 - 2.42T + 89T^{2} \) |
| 97 | \( 1 + 9.84T + 97T^{2} \) |
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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
−7.61954068725610286076560159348, −6.37925107046471853888851944931, −6.21052308157669330895318745564, −5.23613672531484545319124766975, −4.57019704592933715594559036398, −4.14631559332902037174504565753, −2.93409577937183946467022378674, −2.57853564791051116268277129068, −1.52594510143244351257140859070, 0,
1.52594510143244351257140859070, 2.57853564791051116268277129068, 2.93409577937183946467022378674, 4.14631559332902037174504565753, 4.57019704592933715594559036398, 5.23613672531484545319124766975, 6.21052308157669330895318745564, 6.37925107046471853888851944931, 7.61954068725610286076560159348
