| L(s) = 1 | + 2.82·5-s − 14.1·7-s − 20·11-s − 39.5·13-s + 34·17-s + 52·19-s + 62.2·23-s − 117·25-s + 200.·29-s + 110.·31-s − 40.0·35-s + 271.·37-s + 26·41-s + 252·43-s + 345.·47-s − 142.·49-s − 681.·53-s − 56.5·55-s − 364·59-s − 735.·61-s − 112.·65-s + 628·67-s − 333.·71-s + 338·73-s + 282.·77-s + 789.·79-s − 1.03e3·83-s + ⋯ |
| L(s) = 1 | + 0.252·5-s − 0.763·7-s − 0.548·11-s − 0.844·13-s + 0.485·17-s + 0.627·19-s + 0.564·23-s − 0.936·25-s + 1.28·29-s + 0.639·31-s − 0.193·35-s + 1.20·37-s + 0.0990·41-s + 0.893·43-s + 1.07·47-s − 0.416·49-s − 1.76·53-s − 0.138·55-s − 0.803·59-s − 1.54·61-s − 0.213·65-s + 1.14·67-s − 0.557·71-s + 0.541·73-s + 0.418·77-s + 1.12·79-s − 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.82T + 125T^{2} \) |
| 7 | \( 1 + 14.1T + 343T^{2} \) |
| 11 | \( 1 + 20T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 34T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52T + 6.85e3T^{2} \) |
| 23 | \( 1 - 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 26T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252T + 7.95e4T^{2} \) |
| 47 | \( 1 - 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 681.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 364T + 2.05e5T^{2} \) |
| 61 | \( 1 + 735.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 628T + 3.00e5T^{2} \) |
| 71 | \( 1 + 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 338T + 3.89e5T^{2} \) |
| 79 | \( 1 - 789.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 234T + 7.04e5T^{2} \) |
| 97 | \( 1 + 178T + 9.12e5T^{2} \) |
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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
−8.090216327306470616308953274082, −7.60214241522955514365271994062, −6.65903911995254774118057232156, −5.95647068821454249340170427658, −5.12345080342747537723405474849, −4.28611840630863406825542033921, −3.08546346978539821710780490055, −2.54538652406231311701105519025, −1.15254562580518592920482503340, 0,
1.15254562580518592920482503340, 2.54538652406231311701105519025, 3.08546346978539821710780490055, 4.28611840630863406825542033921, 5.12345080342747537723405474849, 5.95647068821454249340170427658, 6.65903911995254774118057232156, 7.60214241522955514365271994062, 8.090216327306470616308953274082
