| L(s) = 1 | − 2-s − 0.181·3-s + 4-s + 3.41·5-s + 0.181·6-s + 3.19·7-s − 8-s − 2.96·9-s − 3.41·10-s − 0.937·11-s − 0.181·12-s + 3.37·13-s − 3.19·14-s − 0.620·15-s + 16-s + 1.87·17-s + 2.96·18-s + 5.76·19-s + 3.41·20-s − 0.580·21-s + 0.937·22-s + 5.78·23-s + 0.181·24-s + 6.69·25-s − 3.37·26-s + 1.08·27-s + 3.19·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.104·3-s + 0.5·4-s + 1.52·5-s + 0.0740·6-s + 1.20·7-s − 0.353·8-s − 0.989·9-s − 1.08·10-s − 0.282·11-s − 0.0523·12-s + 0.934·13-s − 0.855·14-s − 0.160·15-s + 0.250·16-s + 0.454·17-s + 0.699·18-s + 1.32·19-s + 0.764·20-s − 0.126·21-s + 0.199·22-s + 1.20·23-s + 0.0370·24-s + 1.33·25-s − 0.660·26-s + 0.208·27-s + 0.604·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.434002269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.434002269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 0.181T + 3T^{2} \) |
| 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 - 3.19T + 7T^{2} \) |
| 11 | \( 1 + 0.937T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 - 1.87T + 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 - 5.78T + 23T^{2} \) |
| 29 | \( 1 - 4.63T + 29T^{2} \) |
| 37 | \( 1 - 8.37T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 + 7.36T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 + 6.71T + 59T^{2} \) |
| 61 | \( 1 + 8.12T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 8.21T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
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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
−8.167903027536422270351422376732, −7.58357108709491219073073346175, −6.52677388567225796612799717769, −6.01228817123421217012401072950, −5.27544262434185536059283091473, −4.85368628810425305875846472359, −3.25764208392639256209496910086, −2.63980085633414866921549408991, −1.59754081238082770250356268176, −1.02860196861182196865749385528,
1.02860196861182196865749385528, 1.59754081238082770250356268176, 2.63980085633414866921549408991, 3.25764208392639256209496910086, 4.85368628810425305875846472359, 5.27544262434185536059283091473, 6.01228817123421217012401072950, 6.52677388567225796612799717769, 7.58357108709491219073073346175, 8.167903027536422270351422376732
