| L(s) = 1 | − 4.44·5-s − 4.87·7-s + 3.46·11-s + 14.7·25-s + 5.34·29-s + 10.5·31-s + 21.7·35-s + 16.7·49-s − 12.4·53-s − 15.4·55-s − 11.3·59-s − 9.79·73-s − 16.8·77-s + 3.32·79-s − 17.3·83-s − 2·97-s − 11.5·101-s + 20.2·103-s − 11.3·107-s + ⋯ |
| L(s) = 1 | − 1.98·5-s − 1.84·7-s + 1.04·11-s + 2.95·25-s + 0.993·29-s + 1.89·31-s + 3.66·35-s + 2.39·49-s − 1.71·53-s − 2.07·55-s − 1.47·59-s − 1.14·73-s − 1.92·77-s + 0.373·79-s − 1.90·83-s − 0.203·97-s − 1.14·101-s + 1.99·103-s − 1.09·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 4.44T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.34T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 - 3.32T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.988967483278793011697663496352, −7.10625240620324144492886040108, −6.65349862015332969025726023516, −6.05235057756333677023197321143, −4.59896907418805351579380447249, −4.17697399318567713829293741196, −3.23957931095798611527466420214, −2.97577267528829505546894701006, −0.987705560611143327129337532796, 0,
0.987705560611143327129337532796, 2.97577267528829505546894701006, 3.23957931095798611527466420214, 4.17697399318567713829293741196, 4.59896907418805351579380447249, 6.05235057756333677023197321143, 6.65349862015332969025726023516, 7.10625240620324144492886040108, 7.988967483278793011697663496352
