| L(s) = 1 | − 2.69·2-s − 3-s + 5.24·4-s + 1.04·5-s + 2.69·6-s − 0.554·7-s − 8.74·8-s + 9-s − 2.82·10-s − 2.91·11-s − 5.24·12-s + 1.49·14-s − 1.04·15-s + 13.0·16-s − 4.85·17-s − 2.69·18-s + 0.753·19-s + 5.50·20-s + 0.554·21-s + 7.83·22-s + 5.76·23-s + 8.74·24-s − 3.89·25-s − 27-s − 2.91·28-s − 1.91·29-s + 2.82·30-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 0.577·3-s + 2.62·4-s + 0.469·5-s + 1.09·6-s − 0.209·7-s − 3.09·8-s + 0.333·9-s − 0.892·10-s − 0.877·11-s − 1.51·12-s + 0.399·14-s − 0.270·15-s + 3.25·16-s − 1.17·17-s − 0.634·18-s + 0.172·19-s + 1.23·20-s + 0.121·21-s + 1.67·22-s + 1.20·23-s + 1.78·24-s − 0.779·25-s − 0.192·27-s − 0.550·28-s − 0.355·29-s + 0.515·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 1.04T + 5T^{2} \) |
| 7 | \( 1 + 0.554T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 - 0.753T + 19T^{2} \) |
| 23 | \( 1 - 5.76T + 23T^{2} \) |
| 29 | \( 1 + 1.91T + 29T^{2} \) |
| 31 | \( 1 - 9.51T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 + 4.91T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 0.753T + 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 - 1.87T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.33T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 + 9.92T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
| 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
−10.24632024213934150814357778698, −9.709876464886236275882495479204, −8.758256292235516698379198558779, −7.955365973166934187778379192542, −6.90245929966587211773087324747, −6.30911588357971738443301632702, −5.07351394520457083353807688653, −2.92134042800006823419670745599, −1.67584530289778924637150065375, 0,
1.67584530289778924637150065375, 2.92134042800006823419670745599, 5.07351394520457083353807688653, 6.30911588357971738443301632702, 6.90245929966587211773087324747, 7.955365973166934187778379192542, 8.758256292235516698379198558779, 9.709876464886236275882495479204, 10.24632024213934150814357778698
