| L(s) = 1 | − 1.03·3-s − 2.90·5-s − 0.294·7-s − 1.93·9-s + 3.78·11-s − 3.15·13-s + 2.99·15-s − 6.61·17-s + 1.57·19-s + 0.304·21-s + 5.53·23-s + 3.42·25-s + 5.09·27-s + 0.673·29-s + 6.77·31-s − 3.91·33-s + 0.856·35-s + 2.01·37-s + 3.26·39-s + 4.28·41-s + 0.897·43-s + 5.61·45-s + 6.08·47-s − 6.91·49-s + 6.82·51-s + 4.70·53-s − 10.9·55-s + ⋯ |
| L(s) = 1 | − 0.596·3-s − 1.29·5-s − 0.111·7-s − 0.644·9-s + 1.14·11-s − 0.876·13-s + 0.774·15-s − 1.60·17-s + 0.361·19-s + 0.0664·21-s + 1.15·23-s + 0.685·25-s + 0.980·27-s + 0.125·29-s + 1.21·31-s − 0.680·33-s + 0.144·35-s + 0.330·37-s + 0.522·39-s + 0.669·41-s + 0.136·43-s + 0.836·45-s + 0.886·47-s − 0.987·49-s + 0.956·51-s + 0.646·53-s − 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 1.03T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 + 0.294T + 7T^{2} \) |
| 11 | \( 1 - 3.78T + 11T^{2} \) |
| 13 | \( 1 + 3.15T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 1.57T + 19T^{2} \) |
| 23 | \( 1 - 5.53T + 23T^{2} \) |
| 29 | \( 1 - 0.673T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 - 2.01T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 - 0.897T + 43T^{2} \) |
| 47 | \( 1 - 6.08T + 47T^{2} \) |
| 53 | \( 1 - 4.70T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 0.579T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 8.03T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 9.03T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 3.10T + 97T^{2} \) |
| 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
−7.65134789470005655093708094773, −6.91318781044171906274872079243, −6.49552801916525089598609478947, −5.57341553615742133963010070652, −4.59271964735781113600764068989, −4.30586522395089232600709049201, −3.25249300072962967340661415062, −2.48367581825186396224905107812, −0.977135235451922692975186942985, 0,
0.977135235451922692975186942985, 2.48367581825186396224905107812, 3.25249300072962967340661415062, 4.30586522395089232600709049201, 4.59271964735781113600764068989, 5.57341553615742133963010070652, 6.49552801916525089598609478947, 6.91318781044171906274872079243, 7.65134789470005655093708094773
