| L(s) = 1 | − 3-s − 2·4-s − 5-s − 7-s + 9-s − 11-s + 2·12-s + 4·13-s + 15-s + 4·16-s + 2·17-s + 2·20-s + 21-s + 2·23-s − 4·25-s − 27-s + 2·28-s + 4·29-s − 31-s + 33-s + 35-s − 2·36-s − 2·37-s − 4·39-s + 9·41-s + 6·43-s + 2·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.577·12-s + 1.10·13-s + 0.258·15-s + 16-s + 0.485·17-s + 0.447·20-s + 0.218·21-s + 0.417·23-s − 4/5·25-s − 0.192·27-s + 0.377·28-s + 0.742·29-s − 0.179·31-s + 0.174·33-s + 0.169·35-s − 1/3·36-s − 0.328·37-s − 0.640·39-s + 1.40·41-s + 0.914·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.519508201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.519508201\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−13.91151938975983, −13.38337550276623, −12.98115130513342, −12.43166923589490, −12.14709066654243, −11.42464527123204, −10.93395139938447, −10.53624000018226, −9.906285946978247, −9.477453883809785, −8.933646865487241, −8.419829217464164, −7.829803979605009, −7.511547654261139, −6.620330817229750, −6.220640196057531, −5.503708692075599, −5.273272607740040, −4.414983262467488, −4.001244318251077, −3.518259978110623, −2.849609425465985, −1.882007686666404, −0.8781994586337023, −0.5863031112687780,
0.5863031112687780, 0.8781994586337023, 1.882007686666404, 2.849609425465985, 3.518259978110623, 4.001244318251077, 4.414983262467488, 5.273272607740040, 5.503708692075599, 6.220640196057531, 6.620330817229750, 7.511547654261139, 7.829803979605009, 8.419829217464164, 8.933646865487241, 9.477453883809785, 9.906285946978247, 10.53624000018226, 10.93395139938447, 11.42464527123204, 12.14709066654243, 12.43166923589490, 12.98115130513342, 13.38337550276623, 13.91151938975983
