| L(s) = 1 | + 1.53·2-s + 0.364·4-s + 1.68·7-s − 2.51·8-s + 2.97·11-s + 0.232·13-s + 2.59·14-s − 4.59·16-s − 7.45·17-s + 0.753·19-s + 4.57·22-s − 0.872·23-s + 0.358·26-s + 0.614·28-s − 6.87·29-s − 9.81·31-s − 2.03·32-s − 11.4·34-s + 10.1·37-s + 1.15·38-s − 3.79·41-s − 5.27·43-s + 1.08·44-s − 1.34·46-s + 8.56·47-s − 4.15·49-s + 0.0848·52-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.182·4-s + 0.637·7-s − 0.889·8-s + 0.897·11-s + 0.0645·13-s + 0.692·14-s − 1.14·16-s − 1.80·17-s + 0.172·19-s + 0.976·22-s − 0.181·23-s + 0.0702·26-s + 0.116·28-s − 1.27·29-s − 1.76·31-s − 0.360·32-s − 1.96·34-s + 1.66·37-s + 0.187·38-s − 0.593·41-s − 0.804·43-s + 0.163·44-s − 0.197·46-s + 1.24·47-s − 0.593·49-s + 0.0117·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 - 0.232T + 13T^{2} \) |
| 17 | \( 1 + 7.45T + 17T^{2} \) |
| 19 | \( 1 - 0.753T + 19T^{2} \) |
| 23 | \( 1 + 0.872T + 23T^{2} \) |
| 29 | \( 1 + 6.87T + 29T^{2} \) |
| 31 | \( 1 + 9.81T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 3.79T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 - 8.56T + 47T^{2} \) |
| 53 | \( 1 + 5.97T + 53T^{2} \) |
| 59 | \( 1 - 3.85T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 - 1.79T + 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 + 4.34T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.47T + 97T^{2} \) |
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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
−7.65171641900244351823412711568, −6.86317865315394100572566624007, −6.20634993719500030134133342820, −5.51805665107943121205638053782, −4.74692615616547034025448499067, −4.14508154128809990778415477163, −3.57965371226496158603345478186, −2.47134829299393649601629140329, −1.61362647364368332957880704689, 0,
1.61362647364368332957880704689, 2.47134829299393649601629140329, 3.57965371226496158603345478186, 4.14508154128809990778415477163, 4.74692615616547034025448499067, 5.51805665107943121205638053782, 6.20634993719500030134133342820, 6.86317865315394100572566624007, 7.65171641900244351823412711568
