| L(s) = 1 | − 1.86·2-s + 1.47·3-s + 1.48·4-s − 2.74·6-s − 7-s + 0.969·8-s − 0.831·9-s + 2.38·11-s + 2.17·12-s − 5.15·13-s + 1.86·14-s − 4.76·16-s + 3.94·17-s + 1.55·18-s − 8.03·19-s − 1.47·21-s − 4.45·22-s − 23-s + 1.42·24-s + 9.61·26-s − 5.64·27-s − 1.48·28-s − 3.71·29-s + 4.20·31-s + 6.95·32-s + 3.51·33-s − 7.36·34-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.850·3-s + 0.740·4-s − 1.12·6-s − 0.377·7-s + 0.342·8-s − 0.277·9-s + 0.720·11-s + 0.629·12-s − 1.42·13-s + 0.498·14-s − 1.19·16-s + 0.957·17-s + 0.365·18-s − 1.84·19-s − 0.321·21-s − 0.950·22-s − 0.208·23-s + 0.291·24-s + 1.88·26-s − 1.08·27-s − 0.279·28-s − 0.689·29-s + 0.754·31-s + 1.23·32-s + 0.612·33-s − 1.26·34-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
| \(L(1)\) |
\(\approx\) |
\(0.8509006397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8509006397\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.86T + 2T^{2} \) |
| 3 | \( 1 - 1.47T + 3T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 3.94T + 17T^{2} \) |
| 19 | \( 1 + 8.03T + 19T^{2} \) |
| 29 | \( 1 + 3.71T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 + 2.12T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 43 | \( 1 - 5.36T + 43T^{2} \) |
| 47 | \( 1 - 6.57T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 2.07T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 - 5.56T + 67T^{2} \) |
| 71 | \( 1 - 6.20T + 71T^{2} \) |
| 73 | \( 1 - 5.19T + 73T^{2} \) |
| 79 | \( 1 + 6.41T + 79T^{2} \) |
| 83 | \( 1 + 6.39T + 83T^{2} \) |
| 89 | \( 1 + 1.93T + 89T^{2} \) |
| 97 | \( 1 - 5.62T + 97T^{2} \) |
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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
−8.771270654635798580950930233048, −7.81710094951448010843518452338, −7.37195640182321455819986408701, −6.55593927939568523025934756540, −5.63652085092584102679456364851, −4.48562733268522290932760041495, −3.72599612070414807636693384713, −2.55474766622056937735831555205, −2.00814114468363002343117534816, −0.59712263878813369107435089835,
0.59712263878813369107435089835, 2.00814114468363002343117534816, 2.55474766622056937735831555205, 3.72599612070414807636693384713, 4.48562733268522290932760041495, 5.63652085092584102679456364851, 6.55593927939568523025934756540, 7.37195640182321455819986408701, 7.81710094951448010843518452338, 8.771270654635798580950930233048
