| L(s) = 1 | + 2·2-s + 0.871·3-s + 4·4-s − 6.15·5-s + 1.74·6-s + 8·8-s − 26.2·9-s − 12.3·10-s + 25.5·11-s + 3.48·12-s − 47.4·13-s − 5.36·15-s + 16·16-s + 23.8·17-s − 52.4·18-s + 97.0·19-s − 24.6·20-s + 51.1·22-s + 23·23-s + 6.97·24-s − 87.0·25-s − 94.9·26-s − 46.4·27-s − 112.·29-s − 10.7·30-s + 278.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.167·3-s + 0.5·4-s − 0.550·5-s + 0.118·6-s + 0.353·8-s − 0.971·9-s − 0.389·10-s + 0.700·11-s + 0.0838·12-s − 1.01·13-s − 0.0923·15-s + 0.250·16-s + 0.340·17-s − 0.687·18-s + 1.17·19-s − 0.275·20-s + 0.495·22-s + 0.208·23-s + 0.0593·24-s − 0.696·25-s − 0.716·26-s − 0.330·27-s − 0.720·29-s − 0.0653·30-s + 1.61·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.922283551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.922283551\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 0.871T + 27T^{2} \) |
| 5 | \( 1 + 6.15T + 125T^{2} \) |
| 11 | \( 1 - 25.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 47.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 97.0T + 6.85e3T^{2} \) |
| 29 | \( 1 + 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 278.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 125.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 1.95T + 6.89e4T^{2} \) |
| 43 | \( 1 + 185.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 277.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 296.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 384.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 353.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 674.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 769.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 258.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 361.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 977.T + 9.12e5T^{2} \) |
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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
−8.544550021405872174011766849892, −7.79658266550563981475707677066, −7.14807284641892223979929251987, −6.24978012513086123973951121043, −5.41042482328200710100651017592, −4.71322091402872781893673944173, −3.67417347112869556321606480442, −3.07036221545974912637559226865, −2.04811468182549762926941336420, −0.66729983869411878823686439242,
0.66729983869411878823686439242, 2.04811468182549762926941336420, 3.07036221545974912637559226865, 3.67417347112869556321606480442, 4.71322091402872781893673944173, 5.41042482328200710100651017592, 6.24978012513086123973951121043, 7.14807284641892223979929251987, 7.79658266550563981475707677066, 8.544550021405872174011766849892
