| L(s) = 1 | − 1.79·2-s + 1.20·4-s − 5-s + 7-s + 1.41·8-s + 1.79·10-s − 5·11-s − 4.58·13-s − 1.79·14-s − 4.95·16-s + 3·17-s + 3.58·19-s − 1.20·20-s + 8.95·22-s + 4·23-s + 25-s + 8.20·26-s + 1.20·28-s − 29-s + 4·31-s + 6.04·32-s − 5.37·34-s − 35-s − 4·37-s − 6.41·38-s − 1.41·40-s + 9.16·41-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.604·4-s − 0.447·5-s + 0.377·7-s + 0.501·8-s + 0.566·10-s − 1.50·11-s − 1.27·13-s − 0.478·14-s − 1.23·16-s + 0.727·17-s + 0.821·19-s − 0.270·20-s + 1.90·22-s + 0.834·23-s + 0.200·25-s + 1.60·26-s + 0.228·28-s − 0.185·29-s + 0.718·31-s + 1.06·32-s − 0.921·34-s − 0.169·35-s − 0.657·37-s − 1.04·38-s − 0.224·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5778213804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5778213804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 4.58T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 9.16T + 41T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.417T + 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 4.16T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 7.58T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 1.41T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−9.859305826106954012877599441931, −8.767952699352663135566731424857, −7.903555847505768846152238391293, −7.66886914070631679152925175304, −6.79452836761544173571285866947, −5.19039359417866767465910634504, −4.84005163502576090222198072988, −3.25117926876776484171041075535, −2.14483649046238086318972586305, −0.66149352268979187435367943342,
0.66149352268979187435367943342, 2.14483649046238086318972586305, 3.25117926876776484171041075535, 4.84005163502576090222198072988, 5.19039359417866767465910634504, 6.79452836761544173571285866947, 7.66886914070631679152925175304, 7.903555847505768846152238391293, 8.767952699352663135566731424857, 9.859305826106954012877599441931
