| L(s) = 1 | + 3-s − 1.80·5-s − 3.41·7-s + 9-s − 1.92·11-s + 5.29·13-s − 1.80·15-s − 0.758·17-s + 4.73·19-s − 3.41·21-s − 1.72·25-s + 27-s + 4.26·29-s − 3.42·31-s − 1.92·33-s + 6.16·35-s − 3.67·37-s + 5.29·39-s − 7.03·41-s − 3.67·43-s − 1.80·45-s + 10.3·47-s + 4.62·49-s − 0.758·51-s + 7.86·53-s + 3.48·55-s + 4.73·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.808·5-s − 1.28·7-s + 0.333·9-s − 0.581·11-s + 1.46·13-s − 0.466·15-s − 0.183·17-s + 1.08·19-s − 0.744·21-s − 0.345·25-s + 0.192·27-s + 0.792·29-s − 0.614·31-s − 0.335·33-s + 1.04·35-s − 0.604·37-s + 0.848·39-s − 1.09·41-s − 0.560·43-s − 0.269·45-s + 1.50·47-s + 0.661·49-s − 0.106·51-s + 1.08·53-s + 0.470·55-s + 0.627·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6348 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 1.80T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 + 0.758T + 17T^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 + 3.67T + 37T^{2} \) |
| 41 | \( 1 + 7.03T + 41T^{2} \) |
| 43 | \( 1 + 3.67T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 7.86T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 6.20T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 - 6.36T + 79T^{2} \) |
| 83 | \( 1 + 9.77T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 7.62T + 97T^{2} \) |
| 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
−7.55124482094083848763912860912, −7.19606732186032472816560136715, −6.28564765076898734374850364210, −5.68078354653765382568674125448, −4.64748623304900087548664879252, −3.60804726355666112073713644399, −3.46984789795118456285411952455, −2.54342742170529488120289423604, −1.24156764053057941394529597463, 0,
1.24156764053057941394529597463, 2.54342742170529488120289423604, 3.46984789795118456285411952455, 3.60804726355666112073713644399, 4.64748623304900087548664879252, 5.68078354653765382568674125448, 6.28564765076898734374850364210, 7.19606732186032472816560136715, 7.55124482094083848763912860912
