| L(s) = 1 | + 2·2-s + 2.25·3-s + 4·4-s − 5.95·5-s + 4.50·6-s + 8·8-s − 21.9·9-s − 11.9·10-s − 11.6·11-s + 9.01·12-s − 12.7·13-s − 13.4·15-s + 16·16-s − 138.·17-s − 43.8·18-s + 135.·19-s − 23.8·20-s − 23.2·22-s − 23·23-s + 18.0·24-s − 89.4·25-s − 25.4·26-s − 110.·27-s + 99.1·29-s − 26.8·30-s + 184.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.433·3-s + 0.5·4-s − 0.532·5-s + 0.306·6-s + 0.353·8-s − 0.811·9-s − 0.376·10-s − 0.319·11-s + 0.216·12-s − 0.271·13-s − 0.231·15-s + 0.250·16-s − 1.97·17-s − 0.574·18-s + 1.63·19-s − 0.266·20-s − 0.225·22-s − 0.208·23-s + 0.153·24-s − 0.715·25-s − 0.191·26-s − 0.785·27-s + 0.635·29-s − 0.163·30-s + 1.07·31-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
| \(L(2)\) |
\(\approx\) |
\(2.917439885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.917439885\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.25T + 27T^{2} \) |
| 5 | \( 1 + 5.95T + 125T^{2} \) |
| 11 | \( 1 + 11.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 138.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 135.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 99.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 51.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 527.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 904.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 509.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 372.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 749.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 215.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 764.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 480.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.600146379146145210879862650451, −7.82544082921907665631240726211, −7.18255177522083634626624964607, −6.24971153735585025704405408888, −5.45724379568241173530970711798, −4.56756728715040471477390137179, −3.82691054775394257489046675428, −2.82684539284651942955859501760, −2.25137108998555624820017474417, −0.64423862469087677384674109108,
0.64423862469087677384674109108, 2.25137108998555624820017474417, 2.82684539284651942955859501760, 3.82691054775394257489046675428, 4.56756728715040471477390137179, 5.45724379568241173530970711798, 6.24971153735585025704405408888, 7.18255177522083634626624964607, 7.82544082921907665631240726211, 8.600146379146145210879862650451
