| L(s) = 1 | − 2.11·2-s + 1.59·3-s + 2.46·4-s − 3.35·6-s − 7-s − 0.972·8-s − 0.470·9-s − 4.34·11-s + 3.91·12-s − 3.94·13-s + 2.11·14-s − 2.86·16-s + 7.21·17-s + 0.993·18-s + 5.00·19-s − 1.59·21-s + 9.16·22-s − 23-s − 1.54·24-s + 8.33·26-s − 5.51·27-s − 2.46·28-s + 9.13·29-s + 4.73·31-s + 8.00·32-s − 6.90·33-s − 15.2·34-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 0.918·3-s + 1.23·4-s − 1.37·6-s − 0.377·7-s − 0.343·8-s − 0.156·9-s − 1.30·11-s + 1.12·12-s − 1.09·13-s + 0.564·14-s − 0.716·16-s + 1.75·17-s + 0.234·18-s + 1.14·19-s − 0.347·21-s + 1.95·22-s − 0.208·23-s − 0.315·24-s + 1.63·26-s − 1.06·27-s − 0.464·28-s + 1.69·29-s + 0.850·31-s + 1.41·32-s − 1.20·33-s − 2.61·34-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
| \(L(1)\) |
\(\approx\) |
\(0.8506452880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8506452880\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.11T + 2T^{2} \) |
| 3 | \( 1 - 1.59T + 3T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 + 3.94T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 29 | \( 1 - 9.13T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 + 9.97T + 37T^{2} \) |
| 41 | \( 1 - 0.0577T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 + 7.57T + 53T^{2} \) |
| 59 | \( 1 + 2.38T + 59T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 + 0.144T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 - 7.74T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 0.860T + 83T^{2} \) |
| 89 | \( 1 - 4.06T + 89T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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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
−8.232938360013190692394456147299, −8.012501989104498274617464930320, −7.45108969503219059826062466537, −6.64161749599300103124554017820, −5.46915321020470872849027032799, −4.81469978847672278766298921186, −3.23746596315447338685076617138, −2.87891691973466452107633984325, −1.88164811995829447766251442733, −0.61558489368950416212788053315,
0.61558489368950416212788053315, 1.88164811995829447766251442733, 2.87891691973466452107633984325, 3.23746596315447338685076617138, 4.81469978847672278766298921186, 5.46915321020470872849027032799, 6.64161749599300103124554017820, 7.45108969503219059826062466537, 8.012501989104498274617464930320, 8.232938360013190692394456147299
