| L(s) = 1 | − 0.523·2-s − 1.72·4-s − 3.20·7-s + 1.95·8-s − 2.20·11-s + 3.04·13-s + 1.67·14-s + 2.42·16-s + 19-s + 1.15·22-s − 3.24·23-s − 1.59·26-s + 5.52·28-s − 3·29-s − 1.24·31-s − 5.17·32-s − 3.45·37-s − 0.523·38-s − 8.55·41-s + 8·43-s + 3.79·44-s + 1.70·46-s − 3.35·47-s + 3.24·49-s − 5.25·52-s − 0.904·53-s − 6.24·56-s + ⋯ |
| L(s) = 1 | − 0.370·2-s − 0.862·4-s − 1.21·7-s + 0.690·8-s − 0.663·11-s + 0.845·13-s + 0.448·14-s + 0.607·16-s + 0.229·19-s + 0.245·22-s − 0.677·23-s − 0.313·26-s + 1.04·28-s − 0.557·29-s − 0.224·31-s − 0.915·32-s − 0.567·37-s − 0.0850·38-s − 1.33·41-s + 1.21·43-s + 0.572·44-s + 0.251·46-s − 0.489·47-s + 0.464·49-s − 0.729·52-s − 0.124·53-s − 0.835·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6543575175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6543575175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.523T + 2T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 - 3.04T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 + 3.45T + 37T^{2} \) |
| 41 | \( 1 + 8.55T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 3.35T + 47T^{2} \) |
| 53 | \( 1 + 0.904T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 3.40T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 + 7.29T + 71T^{2} \) |
| 73 | \( 1 - 1.09T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 8.20T + 83T^{2} \) |
| 89 | \( 1 - 7.40T + 89T^{2} \) |
| 97 | \( 1 + 7.14T + 97T^{2} \) |
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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
−8.417944399256104180728737582969, −7.82801274537417909261537952875, −7.00345327979905034520097320476, −6.13242721381992165334103546672, −5.51133109641883368493199763228, −4.61063817894385658884689351091, −3.69345072926854569084179204037, −3.14487669859085003760095698092, −1.79882631390591759779819916561, −0.48102949179262398139614569860,
0.48102949179262398139614569860, 1.79882631390591759779819916561, 3.14487669859085003760095698092, 3.69345072926854569084179204037, 4.61063817894385658884689351091, 5.51133109641883368493199763228, 6.13242721381992165334103546672, 7.00345327979905034520097320476, 7.82801274537417909261537952875, 8.417944399256104180728737582969
