L(s) = 1 | + 0.921·2-s + 3-s − 1.15·4-s − 1.44·5-s + 0.921·6-s − 0.134·7-s − 2.90·8-s + 9-s − 1.33·10-s + 3.30·11-s − 1.15·12-s + 13-s − 0.123·14-s − 1.44·15-s − 0.370·16-s − 0.860·17-s + 0.921·18-s − 3.82·19-s + 1.66·20-s − 0.134·21-s + 3.04·22-s + 8.84·23-s − 2.90·24-s − 2.90·25-s + 0.921·26-s + 27-s + 0.154·28-s + ⋯ |
L(s) = 1 | + 0.651·2-s + 0.577·3-s − 0.575·4-s − 0.646·5-s + 0.376·6-s − 0.0508·7-s − 1.02·8-s + 0.333·9-s − 0.421·10-s + 0.997·11-s − 0.332·12-s + 0.277·13-s − 0.0331·14-s − 0.373·15-s − 0.0926·16-s − 0.208·17-s + 0.217·18-s − 0.877·19-s + 0.372·20-s − 0.0293·21-s + 0.649·22-s + 1.84·23-s − 0.592·24-s − 0.581·25-s + 0.180·26-s + 0.192·27-s + 0.0292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.299576379\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.299576379\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 0.921T + 2T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 + 0.134T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 17 | \( 1 + 0.860T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 - 8.84T + 23T^{2} \) |
| 29 | \( 1 - 1.16T + 29T^{2} \) |
| 31 | \( 1 + 8.77T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 - 8.57T + 41T^{2} \) |
| 43 | \( 1 + 0.325T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 1.45T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 6.73T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 4.94T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.632179438459780720060574079872, −7.73413963889326362926644919541, −6.95421458131001825709351658436, −6.22219875171752222760984510876, −5.29052712023423817117927342232, −4.47710611781428676088100878811, −3.80699384970828411849664348189, −3.35740917436074824716599939813, −2.17620747928529158192460529865, −0.77347616369165406749839573586,
0.77347616369165406749839573586, 2.17620747928529158192460529865, 3.35740917436074824716599939813, 3.80699384970828411849664348189, 4.47710611781428676088100878811, 5.29052712023423817117927342232, 6.22219875171752222760984510876, 6.95421458131001825709351658436, 7.73413963889326362926644919541, 8.632179438459780720060574079872