| L(s) = 1 | + 1.72·2-s + 3-s + 0.987·4-s + 1.72·6-s − 7-s − 1.75·8-s + 9-s + 1.78·11-s + 0.987·12-s − 1.28·13-s − 1.72·14-s − 4.99·16-s − 17-s + 1.72·18-s − 0.951·19-s − 21-s + 3.09·22-s − 5.16·23-s − 1.75·24-s − 2.22·26-s + 27-s − 0.987·28-s − 4.09·29-s + 10.8·31-s − 5.14·32-s + 1.78·33-s − 1.72·34-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.493·4-s + 0.705·6-s − 0.377·7-s − 0.618·8-s + 0.333·9-s + 0.539·11-s + 0.284·12-s − 0.356·13-s − 0.461·14-s − 1.24·16-s − 0.242·17-s + 0.407·18-s − 0.218·19-s − 0.218·21-s + 0.659·22-s − 1.07·23-s − 0.357·24-s − 0.435·26-s + 0.192·27-s − 0.186·28-s − 0.760·29-s + 1.94·31-s − 0.908·32-s + 0.311·33-s − 0.296·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 1.72T + 2T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 19 | \( 1 + 0.951T + 19T^{2} \) |
| 23 | \( 1 + 5.16T + 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 5.47T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 - 5.00T + 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 - 0.824T + 53T^{2} \) |
| 59 | \( 1 + 8.91T + 59T^{2} \) |
| 61 | \( 1 - 0.615T + 61T^{2} \) |
| 67 | \( 1 + 2.52T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 0.406T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 9.31T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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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
−7.22709305243418698388367079664, −6.49161580225593886160971798741, −6.04558425870152905393771756333, −5.22432897483907840111511617556, −4.29501336884083020609417170984, −4.11199577036879346911113536562, −3.08898043450567527647080904676, −2.62103837196931493773360296520, −1.55674857376635966149479782348, 0,
1.55674857376635966149479782348, 2.62103837196931493773360296520, 3.08898043450567527647080904676, 4.11199577036879346911113536562, 4.29501336884083020609417170984, 5.22432897483907840111511617556, 6.04558425870152905393771756333, 6.49161580225593886160971798741, 7.22709305243418698388367079664
