| L(s) = 1 | − 2-s + 3-s + 4-s + 3.07·5-s − 6-s − 3.35·7-s − 8-s + 9-s − 3.07·10-s − 2.24·11-s + 12-s + 5.51·13-s + 3.35·14-s + 3.07·15-s + 16-s + 17-s − 18-s − 3.56·19-s + 3.07·20-s − 3.35·21-s + 2.24·22-s − 1.79·23-s − 24-s + 4.47·25-s − 5.51·26-s + 27-s − 3.35·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.37·5-s − 0.408·6-s − 1.26·7-s − 0.353·8-s + 0.333·9-s − 0.973·10-s − 0.676·11-s + 0.288·12-s + 1.52·13-s + 0.896·14-s + 0.794·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 0.818·19-s + 0.688·20-s − 0.732·21-s + 0.478·22-s − 0.373·23-s − 0.204·24-s + 0.894·25-s − 1.08·26-s + 0.192·27-s − 0.633·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 - 0.808T + 29T^{2} \) |
| 31 | \( 1 + 9.38T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 4.07T + 53T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 9.62T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 + 8.25T + 79T^{2} \) |
| 83 | \( 1 - 4.41T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 + 7.53T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.993012745526995781524240382114, −6.79551537269163334605808392216, −6.49415101312963904043016373811, −5.83123406857818351668415619858, −5.03974793092876934441540792858, −3.56706119390399283398347880614, −3.22280248212781968533346370892, −2.09453005606232918560906226706, −1.55806017101501346671783268287, 0,
1.55806017101501346671783268287, 2.09453005606232918560906226706, 3.22280248212781968533346370892, 3.56706119390399283398347880614, 5.03974793092876934441540792858, 5.83123406857818351668415619858, 6.49415101312963904043016373811, 6.79551537269163334605808392216, 7.993012745526995781524240382114
