| L(s) = 1 | + 3-s − 3.23·5-s + 1.23·7-s + 9-s − 11-s − 0.763·13-s − 3.23·15-s + 4.47·17-s − 2.47·19-s + 1.23·21-s − 7.70·23-s + 5.47·25-s + 27-s + 0.472·29-s − 33-s − 4.00·35-s − 6·37-s − 0.763·39-s − 6.94·41-s + 4.94·43-s − 3.23·45-s + 7.70·47-s − 5.47·49-s + 4.47·51-s − 11.2·53-s + 3.23·55-s − 2.47·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.44·5-s + 0.467·7-s + 0.333·9-s − 0.301·11-s − 0.211·13-s − 0.835·15-s + 1.08·17-s − 0.567·19-s + 0.269·21-s − 1.60·23-s + 1.09·25-s + 0.192·27-s + 0.0876·29-s − 0.174·33-s − 0.676·35-s − 0.986·37-s − 0.122·39-s − 1.08·41-s + 0.753·43-s − 0.482·45-s + 1.12·47-s − 0.781·49-s + 0.626·51-s − 1.54·53-s + 0.436·55-s − 0.327·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 13 | \( 1 + 0.763T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 - 7.70T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 7.70T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−8.427712910662593196767753209642, −7.923768885675984494067677302613, −7.53660103765606559156664779400, −6.51270608098609688458266922393, −5.37861847386452743979255689842, −4.42357508473348319009436717819, −3.77996724084366094612819246393, −2.91933875363738625831096412376, −1.64275786369660668880662440320, 0,
1.64275786369660668880662440320, 2.91933875363738625831096412376, 3.77996724084366094612819246393, 4.42357508473348319009436717819, 5.37861847386452743979255689842, 6.51270608098609688458266922393, 7.53660103765606559156664779400, 7.923768885675984494067677302613, 8.427712910662593196767753209642
