| L(s) = 1 | + 2-s + 2.68·3-s + 4-s + 2.52·5-s + 2.68·6-s − 7-s + 8-s + 4.21·9-s + 2.52·10-s + 3.41·11-s + 2.68·12-s − 0.744·13-s − 14-s + 6.78·15-s + 16-s − 0.526·17-s + 4.21·18-s + 7.37·19-s + 2.52·20-s − 2.68·21-s + 3.41·22-s + 2.68·24-s + 1.38·25-s − 0.744·26-s + 3.25·27-s − 28-s − 4.92·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.55·3-s + 0.5·4-s + 1.12·5-s + 1.09·6-s − 0.377·7-s + 0.353·8-s + 1.40·9-s + 0.798·10-s + 1.02·11-s + 0.775·12-s − 0.206·13-s − 0.267·14-s + 1.75·15-s + 0.250·16-s − 0.127·17-s + 0.992·18-s + 1.69·19-s + 0.564·20-s − 0.586·21-s + 0.727·22-s + 0.548·24-s + 0.276·25-s − 0.146·26-s + 0.626·27-s − 0.188·28-s − 0.914·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.745449929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.745449929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 13 | \( 1 + 0.744T + 13T^{2} \) |
| 17 | \( 1 + 0.526T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 - 2.18T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 - 6.27T + 53T^{2} \) |
| 59 | \( 1 - 7.73T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 0.603T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 0.631T + 79T^{2} \) |
| 83 | \( 1 + 4.74T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−7.908012620001624868040763493066, −6.97783345460047255313930799019, −6.76029854505616005329107161133, −5.57580813739842049822075090422, −5.25775588678802209305415321760, −3.89443902838383760838681876282, −3.62634617984024371005067912803, −2.73893119531073135994809070274, −2.05084942454454073425382699537, −1.33171725538161297199966724012,
1.33171725538161297199966724012, 2.05084942454454073425382699537, 2.73893119531073135994809070274, 3.62634617984024371005067912803, 3.89443902838383760838681876282, 5.25775588678802209305415321760, 5.57580813739842049822075090422, 6.76029854505616005329107161133, 6.97783345460047255313930799019, 7.908012620001624868040763493066
