| L(s) = 1 | + 1.61·2-s + 0.618·4-s + 0.874·5-s − 2.23·8-s + 1.41·10-s + 11-s + 1.74·13-s − 4.85·16-s − 3.16·17-s − 5.45·19-s + 0.540·20-s + 1.61·22-s − 2.76·23-s − 4.23·25-s + 2.82·26-s − 3.23·29-s − 0.333·31-s − 3.38·32-s − 5.11·34-s + 2.70·37-s − 8.81·38-s − 1.95·40-s − 10.0·41-s − 11.9·43-s + 0.618·44-s − 4.47·46-s + 12.5·47-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s + 0.390·5-s − 0.790·8-s + 0.447·10-s + 0.301·11-s + 0.484·13-s − 1.21·16-s − 0.766·17-s − 1.25·19-s + 0.120·20-s + 0.344·22-s − 0.576·23-s − 0.847·25-s + 0.554·26-s − 0.600·29-s − 0.0599·31-s − 0.597·32-s − 0.877·34-s + 0.445·37-s − 1.43·38-s − 0.309·40-s − 1.56·41-s − 1.82·43-s + 0.0931·44-s − 0.659·46-s + 1.82·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - 0.874T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 + 3.16T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 + 3.23T + 29T^{2} \) |
| 31 | \( 1 + 0.333T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 - 8.15T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 7.47T + 67T^{2} \) |
| 71 | \( 1 + 3.47T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 7.19T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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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
−8.268122768508118862326063593244, −6.98305650044401627287749585359, −6.43021687448417899479212303884, −5.76971433271049052774411885271, −5.09419432305918916752547724142, −4.11744463512663093955185739729, −3.78452404676519489992536788313, −2.60440045520731440966278171180, −1.78116947301212301719726055390, 0,
1.78116947301212301719726055390, 2.60440045520731440966278171180, 3.78452404676519489992536788313, 4.11744463512663093955185739729, 5.09419432305918916752547724142, 5.76971433271049052774411885271, 6.43021687448417899479212303884, 6.98305650044401627287749585359, 8.268122768508118862326063593244
