| L(s) = 1 | − 1.61·3-s + 4.71·7-s − 0.381·9-s − 5.95·11-s + 4.91·13-s − 4.10·17-s − 8.53·19-s − 7.63·21-s + 3.48·23-s + 5.47·27-s − 0.763·29-s + 4.47·31-s + 9.63·33-s + 1.55·37-s − 7.95·39-s + 5.01·41-s − 1.37·43-s + 5.63·47-s + 15.2·49-s + 6.63·51-s − 0.472·53-s + 13.8·57-s + 0.662·59-s + 6.19·61-s − 1.80·63-s − 11.2·67-s − 5.63·69-s + ⋯ |
| L(s) = 1 | − 0.934·3-s + 1.78·7-s − 0.127·9-s − 1.79·11-s + 1.36·13-s − 0.994·17-s − 1.95·19-s − 1.66·21-s + 0.726·23-s + 1.05·27-s − 0.141·29-s + 0.803·31-s + 1.67·33-s + 0.255·37-s − 1.27·39-s + 0.783·41-s − 0.209·43-s + 0.822·47-s + 2.18·49-s + 0.929·51-s − 0.0648·53-s + 1.82·57-s + 0.0862·59-s + 0.793·61-s − 0.227·63-s − 1.37·67-s − 0.678·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 + 5.95T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 + 4.10T + 17T^{2} \) |
| 19 | \( 1 + 8.53T + 19T^{2} \) |
| 23 | \( 1 - 3.48T + 23T^{2} \) |
| 29 | \( 1 + 0.763T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 + 1.37T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 0.662T + 59T^{2} \) |
| 61 | \( 1 - 6.19T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 - 3.40T + 73T^{2} \) |
| 79 | \( 1 + 4.91T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
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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
−7.32349973055718805928525864061, −6.49937886256524222154948226413, −5.81385198066731593577571091620, −5.33192571880601818988694783438, −4.56530871161344651718902095636, −4.23757955026529955856155629988, −2.78328511275617759694178860387, −2.13725468986192389147395190118, −1.11936970966046570984219794831, 0,
1.11936970966046570984219794831, 2.13725468986192389147395190118, 2.78328511275617759694178860387, 4.23757955026529955856155629988, 4.56530871161344651718902095636, 5.33192571880601818988694783438, 5.81385198066731593577571091620, 6.49937886256524222154948226413, 7.32349973055718805928525864061
