| L(s) = 1 | − 2.25·2-s − 0.777·3-s + 3.09·4-s + 1.75·6-s − 0.123·7-s − 2.48·8-s − 2.39·9-s − 2.40·12-s − 5.49·13-s + 0.279·14-s − 0.596·16-s − 0.519·17-s + 5.40·18-s + 3.15·19-s + 0.0963·21-s + 7.92·23-s + 1.92·24-s + 12.4·26-s + 4.19·27-s − 0.384·28-s − 4.07·29-s + 7.04·31-s + 6.30·32-s + 1.17·34-s − 7.42·36-s + 8.78·37-s − 7.13·38-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.448·3-s + 1.54·4-s + 0.716·6-s − 0.0468·7-s − 0.876·8-s − 0.798·9-s − 0.695·12-s − 1.52·13-s + 0.0748·14-s − 0.149·16-s − 0.125·17-s + 1.27·18-s + 0.724·19-s + 0.0210·21-s + 1.65·23-s + 0.393·24-s + 2.43·26-s + 0.807·27-s − 0.0725·28-s − 0.757·29-s + 1.26·31-s + 1.11·32-s + 0.201·34-s − 1.23·36-s + 1.44·37-s − 1.15·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.25T + 2T^{2} \) |
| 3 | \( 1 + 0.777T + 3T^{2} \) |
| 7 | \( 1 + 0.123T + 7T^{2} \) |
| 13 | \( 1 + 5.49T + 13T^{2} \) |
| 17 | \( 1 + 0.519T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 - 7.92T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 - 8.78T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 + 3.42T + 43T^{2} \) |
| 47 | \( 1 + 0.456T + 47T^{2} \) |
| 53 | \( 1 + 0.0354T + 53T^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 - 7.68T + 61T^{2} \) |
| 67 | \( 1 - 2.53T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 8.52T + 73T^{2} \) |
| 79 | \( 1 - 6.27T + 79T^{2} \) |
| 83 | \( 1 + 0.626T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 1.83T + 97T^{2} \) |
| 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
−8.377186975575240076699309430336, −7.76492530943127561751871098873, −7.01413705185360436639769359293, −6.43513748409708196044495868951, −5.30317004373139834678407365927, −4.70928733263398511268318121431, −3.09925626818282474702696627526, −2.36792514318448067927864423377, −1.06077041459891431252505156392, 0,
1.06077041459891431252505156392, 2.36792514318448067927864423377, 3.09925626818282474702696627526, 4.70928733263398511268318121431, 5.30317004373139834678407365927, 6.43513748409708196044495868951, 7.01413705185360436639769359293, 7.76492530943127561751871098873, 8.377186975575240076699309430336
