| L(s) = 1 | − 3.21·3-s − 2.32·5-s − 7-s + 7.33·9-s − 11-s − 13-s + 7.46·15-s + 7.61·17-s + 7.80·19-s + 3.21·21-s − 9.32·23-s + 0.393·25-s − 13.9·27-s + 3.30·29-s − 9.29·31-s + 3.21·33-s + 2.32·35-s + 3.92·37-s + 3.21·39-s − 10.6·41-s + 7.32·43-s − 17.0·45-s − 10.8·47-s + 49-s − 24.4·51-s − 0.263·53-s + 2.32·55-s + ⋯ |
| L(s) = 1 | − 1.85·3-s − 1.03·5-s − 0.377·7-s + 2.44·9-s − 0.301·11-s − 0.277·13-s + 1.92·15-s + 1.84·17-s + 1.78·19-s + 0.701·21-s − 1.94·23-s + 0.0786·25-s − 2.68·27-s + 0.613·29-s − 1.66·31-s + 0.559·33-s + 0.392·35-s + 0.645·37-s + 0.514·39-s − 1.66·41-s + 1.11·43-s − 2.53·45-s − 1.58·47-s + 0.142·49-s − 3.42·51-s − 0.0361·53-s + 0.313·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4573750648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4573750648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.21T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 17 | \( 1 - 7.61T + 17T^{2} \) |
| 19 | \( 1 - 7.80T + 19T^{2} \) |
| 23 | \( 1 + 9.32T + 23T^{2} \) |
| 29 | \( 1 - 3.30T + 29T^{2} \) |
| 31 | \( 1 + 9.29T + 31T^{2} \) |
| 37 | \( 1 - 3.92T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 0.263T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 + 0.0316T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 - 3.34T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 6.71T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 7.56T + 97T^{2} \) |
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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
−8.015173136827951313840841588020, −7.63871295992185270023506113984, −7.00018673221243394588473219297, −6.04012355619565932836352551464, −5.52104687075762202723765042230, −4.90725683031360722367497098872, −3.92509800379475493951588395028, −3.29375380042648363337391567031, −1.53047492106899188794709707974, −0.44653826345841452861847187969,
0.44653826345841452861847187969, 1.53047492106899188794709707974, 3.29375380042648363337391567031, 3.92509800379475493951588395028, 4.90725683031360722367497098872, 5.52104687075762202723765042230, 6.04012355619565932836352551464, 7.00018673221243394588473219297, 7.63871295992185270023506113984, 8.015173136827951313840841588020
