| L(s) = 1 | + 1.67·2-s − 3.24·3-s + 0.806·4-s − 5.44·6-s − 7-s − 1.99·8-s + 7.54·9-s − 4.71·11-s − 2.62·12-s − 3.32·13-s − 1.67·14-s − 4.96·16-s − 1.29·17-s + 12.6·18-s + 2.45·19-s + 3.24·21-s − 7.90·22-s − 23-s + 6.49·24-s − 5.56·26-s − 14.7·27-s − 0.806·28-s − 5.36·29-s + 5.21·31-s − 4.31·32-s + 15.3·33-s − 2.16·34-s + ⋯ |
| L(s) = 1 | + 1.18·2-s − 1.87·3-s + 0.403·4-s − 2.22·6-s − 0.377·7-s − 0.706·8-s + 2.51·9-s − 1.42·11-s − 0.756·12-s − 0.921·13-s − 0.447·14-s − 1.24·16-s − 0.313·17-s + 2.98·18-s + 0.562·19-s + 0.708·21-s − 1.68·22-s − 0.208·23-s + 1.32·24-s − 1.09·26-s − 2.84·27-s − 0.152·28-s − 0.995·29-s + 0.936·31-s − 0.763·32-s + 2.66·33-s − 0.370·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5732228689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5732228689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 3 | \( 1 + 3.24T + 3T^{2} \) |
| 11 | \( 1 + 4.71T + 11T^{2} \) |
| 13 | \( 1 + 3.32T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 29 | \( 1 + 5.36T + 29T^{2} \) |
| 31 | \( 1 - 5.21T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 9.56T + 41T^{2} \) |
| 43 | \( 1 + 6.36T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 - 8.55T + 59T^{2} \) |
| 61 | \( 1 - 3.86T + 61T^{2} \) |
| 67 | \( 1 - 4.50T + 67T^{2} \) |
| 71 | \( 1 - 9.57T + 71T^{2} \) |
| 73 | \( 1 + 2.46T + 73T^{2} \) |
| 79 | \( 1 + 9.45T + 79T^{2} \) |
| 83 | \( 1 + 6.14T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 5.54T + 97T^{2} \) |
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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
−8.267046944874159082172822910067, −7.16996698481370273312319174010, −6.80266426849458723903307648241, −5.87992234379401450866721454784, −5.27891372035123191353823505099, −5.03780901227596619153394774206, −4.19153995804502943887167099634, −3.22972717665292953754371999528, −2.08732637000396345907650634206, −0.37596248321627975176899671068,
0.37596248321627975176899671068, 2.08732637000396345907650634206, 3.22972717665292953754371999528, 4.19153995804502943887167099634, 5.03780901227596619153394774206, 5.27891372035123191353823505099, 5.87992234379401450866721454784, 6.80266426849458723903307648241, 7.16996698481370273312319174010, 8.267046944874159082172822910067
