| L(s) = 1 | − 2.67·2-s + 1.88·3-s + 5.16·4-s − 5.04·6-s − 1.59·7-s − 8.47·8-s + 0.551·9-s + 2.39·11-s + 9.73·12-s + 0.0275·13-s + 4.26·14-s + 12.3·16-s − 3.49·17-s − 1.47·18-s + 1.43·19-s − 2.99·21-s − 6.41·22-s + 7.45·23-s − 15.9·24-s − 0.0737·26-s − 4.61·27-s − 8.22·28-s + 2.60·29-s + 2.43·31-s − 16.1·32-s + 4.51·33-s + 9.35·34-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 1.08·3-s + 2.58·4-s − 2.05·6-s − 0.601·7-s − 2.99·8-s + 0.183·9-s + 0.722·11-s + 2.81·12-s + 0.00763·13-s + 1.13·14-s + 3.09·16-s − 0.847·17-s − 0.348·18-s + 0.329·19-s − 0.654·21-s − 1.36·22-s + 1.55·23-s − 3.26·24-s − 0.0144·26-s − 0.887·27-s − 1.55·28-s + 0.484·29-s + 0.437·31-s − 2.85·32-s + 0.786·33-s + 1.60·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 3 | \( 1 - 1.88T + 3T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 11 | \( 1 - 2.39T + 11T^{2} \) |
| 13 | \( 1 - 0.0275T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 - 2.60T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 + 8.17T + 37T^{2} \) |
| 41 | \( 1 + 7.97T + 41T^{2} \) |
| 43 | \( 1 - 3.35T + 43T^{2} \) |
| 47 | \( 1 + 4.93T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 - 3.26T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 6.36T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 - 8.60T + 79T^{2} \) |
| 83 | \( 1 - 0.554T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
| 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
−8.020950964814441226842419442600, −7.19202845746646627167858314361, −6.72572176136732737115746319350, −6.09795244958737276499089074704, −4.82507022832463528682120549007, −3.39844579561513943455289264768, −3.04359395170936473507630118510, −2.09818521401782383446358549919, −1.28367901720877722180362349744, 0,
1.28367901720877722180362349744, 2.09818521401782383446358549919, 3.04359395170936473507630118510, 3.39844579561513943455289264768, 4.82507022832463528682120549007, 6.09795244958737276499089074704, 6.72572176136732737115746319350, 7.19202845746646627167858314361, 8.020950964814441226842419442600
