| L(s) = 1 | − 5-s + 1.49·7-s − 4.73·11-s − 1.92·13-s − 6.33·17-s + 6.73·19-s − 23-s + 25-s + 1.74·29-s + 6.26·31-s − 1.49·35-s + 9.53·37-s − 2.07·41-s + 8.65·43-s + 6.29·47-s − 4.75·49-s + 8.00·53-s + 4.73·55-s − 0.428·59-s − 4.42·61-s + 1.92·65-s − 12.5·67-s + 7.08·71-s − 3.43·73-s − 7.08·77-s − 13.1·79-s − 11.1·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.565·7-s − 1.42·11-s − 0.534·13-s − 1.53·17-s + 1.54·19-s − 0.208·23-s + 0.200·25-s + 0.323·29-s + 1.12·31-s − 0.253·35-s + 1.56·37-s − 0.324·41-s + 1.32·43-s + 0.918·47-s − 0.679·49-s + 1.09·53-s + 0.638·55-s − 0.0557·59-s − 0.567·61-s + 0.238·65-s − 1.53·67-s + 0.841·71-s − 0.402·73-s − 0.807·77-s − 1.48·79-s − 1.22·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 + 6.33T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 29 | \( 1 - 1.74T + 29T^{2} \) |
| 31 | \( 1 - 6.26T + 31T^{2} \) |
| 37 | \( 1 - 9.53T + 37T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 - 8.65T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 - 8.00T + 53T^{2} \) |
| 59 | \( 1 + 0.428T + 59T^{2} \) |
| 61 | \( 1 + 4.42T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 7.08T + 71T^{2} \) |
| 73 | \( 1 + 3.43T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 8.52T + 89T^{2} \) |
| 97 | \( 1 + 1.71T + 97T^{2} \) |
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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
−7.65656052489929185690526055987, −6.93095597675073115251602425966, −6.01698247051427096509932144706, −5.26568095712908543282029551056, −4.65422910886581111102874654043, −4.08154547894327224074624919613, −2.77773608075863997609405492733, −2.52209972339267570649971364319, −1.16478462520119766240477514849, 0,
1.16478462520119766240477514849, 2.52209972339267570649971364319, 2.77773608075863997609405492733, 4.08154547894327224074624919613, 4.65422910886581111102874654043, 5.26568095712908543282029551056, 6.01698247051427096509932144706, 6.93095597675073115251602425966, 7.65656052489929185690526055987
