| L(s) = 1 | − 3-s + 5-s − 0.688·7-s + 9-s − 2·11-s + 3.73·13-s − 15-s + 2.28·17-s + 0.688·21-s − 2.90·23-s + 25-s − 27-s − 4.02·29-s − 31-s + 2·33-s − 0.688·35-s − 8.79·37-s − 3.73·39-s + 2.75·41-s − 1.05·43-s + 45-s − 4.70·47-s − 6.52·49-s − 2.28·51-s + 2.14·53-s − 2·55-s + 6.44·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.260·7-s + 0.333·9-s − 0.603·11-s + 1.03·13-s − 0.258·15-s + 0.553·17-s + 0.150·21-s − 0.605·23-s + 0.200·25-s − 0.192·27-s − 0.746·29-s − 0.179·31-s + 0.348·33-s − 0.116·35-s − 1.44·37-s − 0.598·39-s + 0.430·41-s − 0.160·43-s + 0.149·45-s − 0.686·47-s − 0.932·49-s − 0.319·51-s + 0.295·53-s − 0.269·55-s + 0.839·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 0.688T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 - 2.28T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 - 2.75T + 41T^{2} \) |
| 43 | \( 1 + 1.05T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 - 2.14T + 53T^{2} \) |
| 59 | \( 1 - 6.44T + 59T^{2} \) |
| 61 | \( 1 + 5.05T + 61T^{2} \) |
| 67 | \( 1 - 5.44T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 + 8.92T + 73T^{2} \) |
| 79 | \( 1 - 8.38T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−7.53333640659822103981268826497, −6.70406668294656445826128028787, −6.10688325980426750387041046262, −5.51118090761272966970292732896, −4.90932484517098327126495648645, −3.87108106333325805092640681357, −3.24780189096509911922560865760, −2.12464417045084418329065396411, −1.26469010717315475648564022406, 0,
1.26469010717315475648564022406, 2.12464417045084418329065396411, 3.24780189096509911922560865760, 3.87108106333325805092640681357, 4.90932484517098327126495648645, 5.51118090761272966970292732896, 6.10688325980426750387041046262, 6.70406668294656445826128028787, 7.53333640659822103981268826497
