| L(s) = 1 | + 3-s − 3.64·5-s − 1.45·7-s + 9-s − 6.42·11-s − 4.46·13-s − 3.64·15-s − 3.08·17-s − 1.23·19-s − 1.45·21-s + 1.06·23-s + 8.24·25-s + 27-s + 1.27·29-s + 4.58·31-s − 6.42·33-s + 5.28·35-s + 3.59·37-s − 4.46·39-s − 9.41·41-s + 1.44·43-s − 3.64·45-s − 3.82·47-s − 4.89·49-s − 3.08·51-s + 9.67·53-s + 23.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.62·5-s − 0.548·7-s + 0.333·9-s − 1.93·11-s − 1.23·13-s − 0.939·15-s − 0.748·17-s − 0.282·19-s − 0.316·21-s + 0.222·23-s + 1.64·25-s + 0.192·27-s + 0.236·29-s + 0.823·31-s − 1.11·33-s + 0.893·35-s + 0.591·37-s − 0.715·39-s − 1.46·41-s + 0.220·43-s − 0.542·45-s − 0.557·47-s − 0.698·49-s − 0.432·51-s + 1.32·53-s + 3.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5861424980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5861424980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 + 6.42T + 11T^{2} \) |
| 13 | \( 1 + 4.46T + 13T^{2} \) |
| 17 | \( 1 + 3.08T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 - 1.27T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 - 3.59T + 37T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 - 1.44T + 43T^{2} \) |
| 47 | \( 1 + 3.82T + 47T^{2} \) |
| 53 | \( 1 - 9.67T + 53T^{2} \) |
| 59 | \( 1 - 7.68T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 + 7.73T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 2.25T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 - 3.71T + 89T^{2} \) |
| 97 | \( 1 - 5.92T + 97T^{2} \) |
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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
−8.281796732235196041658357359032, −7.76660262393015367882605360588, −7.24052667504083940302868414894, −6.50242476221095606715892489477, −5.13630945406012565435634268285, −4.67844470071681583362286748744, −3.76803132922314500768506889386, −2.91608994315836555009535425993, −2.36345706960652270200607565194, −0.39380280619834365659702284329,
0.39380280619834365659702284329, 2.36345706960652270200607565194, 2.91608994315836555009535425993, 3.76803132922314500768506889386, 4.67844470071681583362286748744, 5.13630945406012565435634268285, 6.50242476221095606715892489477, 7.24052667504083940302868414894, 7.76660262393015367882605360588, 8.281796732235196041658357359032
