L(s) = 1 | − 2-s + 4-s − 1.58·5-s − 8-s + 1.58·10-s − 1.58·11-s + 4.81·13-s + 16-s − 5.39·17-s − 7.09·19-s − 1.58·20-s + 1.58·22-s + 0.300·23-s − 2.47·25-s − 4.81·26-s − 8.27·29-s + 2.71·31-s − 32-s + 5.39·34-s − 37-s + 7.09·38-s + 1.58·40-s + 5.87·41-s + 1.66·43-s − 1.58·44-s − 0.300·46-s − 2.66·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.710·5-s − 0.353·8-s + 0.502·10-s − 0.478·11-s + 1.33·13-s + 0.250·16-s − 1.30·17-s − 1.62·19-s − 0.355·20-s + 0.338·22-s + 0.0626·23-s − 0.495·25-s − 0.943·26-s − 1.53·29-s + 0.487·31-s − 0.176·32-s + 0.925·34-s − 0.164·37-s + 1.15·38-s + 0.251·40-s + 0.917·41-s + 0.254·43-s − 0.239·44-s − 0.0442·46-s − 0.388·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6738777162\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6738777162\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 + 5.39T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 0.300T + 23T^{2} \) |
| 29 | \( 1 + 8.27T + 29T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 5.87T + 41T^{2} \) |
| 43 | \( 1 - 1.66T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 + 4.88T + 53T^{2} \) |
| 59 | \( 1 + 6.47T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 8.38T + 79T^{2} \) |
| 83 | \( 1 - 2.36T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 - 1.42T + 97T^{2} \) |
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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
−7.976739312073664393624776983462, −7.32216173161832031576389709228, −6.39665561338245461779020785845, −6.13158804860481656205969570569, −5.00616206240517861273297422338, −4.10187072939272181441175559035, −3.62330281616465740476910839646, −2.46560304129758732137031051691, −1.75434161473599036725381974994, −0.44010847225000101284073930787,
0.44010847225000101284073930787, 1.75434161473599036725381974994, 2.46560304129758732137031051691, 3.62330281616465740476910839646, 4.10187072939272181441175559035, 5.00616206240517861273297422338, 6.13158804860481656205969570569, 6.39665561338245461779020785845, 7.32216173161832031576389709228, 7.976739312073664393624776983462