| L(s) = 1 | − 2.03·2-s + 3.04·3-s + 2.16·4-s − 6.20·6-s − 3.73·7-s − 0.327·8-s + 6.26·9-s − 2.44·11-s + 6.57·12-s − 2.67·13-s + 7.62·14-s − 3.65·16-s + 4.31·17-s − 12.7·18-s + 4.90·19-s − 11.3·21-s + 4.99·22-s − 3.82·23-s − 0.998·24-s + 5.45·26-s + 9.93·27-s − 8.07·28-s − 1.38·29-s + 7.03·31-s + 8.10·32-s − 7.45·33-s − 8.80·34-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.75·3-s + 1.08·4-s − 2.53·6-s − 1.41·7-s − 0.115·8-s + 2.08·9-s − 0.738·11-s + 1.89·12-s − 0.742·13-s + 2.03·14-s − 0.913·16-s + 1.04·17-s − 3.01·18-s + 1.12·19-s − 2.48·21-s + 1.06·22-s − 0.796·23-s − 0.203·24-s + 1.07·26-s + 1.91·27-s − 1.52·28-s − 0.257·29-s + 1.26·31-s + 1.43·32-s − 1.29·33-s − 1.51·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 3 | \( 1 - 3.04T + 3T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 - 4.31T + 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 7.03T + 31T^{2} \) |
| 37 | \( 1 + 9.99T + 37T^{2} \) |
| 41 | \( 1 - 0.473T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + 4.83T + 47T^{2} \) |
| 53 | \( 1 - 4.16T + 53T^{2} \) |
| 59 | \( 1 - 9.97T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 6.32T + 79T^{2} \) |
| 83 | \( 1 + 9.56T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 6.00T + 97T^{2} \) |
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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
−8.012637073009103135388840261173, −7.19576548753621867505400315918, −7.00639847317944988332153375355, −5.76810762879458600212917285465, −4.66570119404445552658967993856, −3.54101992870714566297638494793, −3.02536241704714356694952495838, −2.32698571346562086251136182175, −1.33333074226739113637408499300, 0,
1.33333074226739113637408499300, 2.32698571346562086251136182175, 3.02536241704714356694952495838, 3.54101992870714566297638494793, 4.66570119404445552658967993856, 5.76810762879458600212917285465, 7.00639847317944988332153375355, 7.19576548753621867505400315918, 8.012637073009103135388840261173
