| L(s) = 1 | + 2.74·2-s + 5.53·4-s − 2.53·5-s + 9.70·8-s − 6.95·10-s − 2.42·11-s + 0.421·13-s + 15.5·16-s + 6.53·17-s + 19-s − 14.0·20-s − 6.64·22-s + 1.57·23-s + 1.42·25-s + 1.15·26-s − 6.02·29-s − 3.48·31-s + 23.3·32-s + 17.9·34-s − 2.84·37-s + 2.74·38-s − 24.5·40-s + 8.42·41-s + 11.4·43-s − 13.4·44-s + 4.33·46-s + 8.02·47-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.76·4-s − 1.13·5-s + 3.42·8-s − 2.19·10-s − 0.730·11-s + 0.116·13-s + 3.88·16-s + 1.58·17-s + 0.229·19-s − 3.13·20-s − 1.41·22-s + 0.329·23-s + 0.284·25-s + 0.226·26-s − 1.11·29-s − 0.626·31-s + 4.11·32-s + 3.07·34-s − 0.467·37-s + 0.445·38-s − 3.88·40-s + 1.31·41-s + 1.75·43-s − 2.02·44-s + 0.638·46-s + 1.17·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.833757621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.833757621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 11 | \( 1 + 2.42T + 11T^{2} \) |
| 13 | \( 1 - 0.421T + 13T^{2} \) |
| 17 | \( 1 - 6.53T + 17T^{2} \) |
| 23 | \( 1 - 1.57T + 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 + 3.48T + 31T^{2} \) |
| 37 | \( 1 + 2.84T + 37T^{2} \) |
| 41 | \( 1 - 8.42T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 8.02T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 6.84T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 2.11T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−7.47708445251210595757311926966, −7.22876433159530373931656782767, −6.07754789763155342925578437157, −5.53066161224532061723825873260, −5.07226652089780549461773131078, −3.94508301422335832264238544030, −3.87502723598413478682115124487, −2.97071684773156950027225082560, −2.25630874526661134693189274598, −0.973361950100451716851109431226,
0.973361950100451716851109431226, 2.25630874526661134693189274598, 2.97071684773156950027225082560, 3.87502723598413478682115124487, 3.94508301422335832264238544030, 5.07226652089780549461773131078, 5.53066161224532061723825873260, 6.07754789763155342925578437157, 7.22876433159530373931656782767, 7.47708445251210595757311926966
