| L(s) = 1 | − 2.64·3-s − 1.64·5-s + 4.00·9-s − 11-s + 5·13-s + 4.35·15-s + 6·17-s + 5.64·19-s − 1.64·23-s − 2.29·25-s − 2.64·27-s + 6.29·29-s + 4·31-s + 2.64·33-s + 3.64·37-s − 13.2·39-s + 10.9·41-s + 4·43-s − 6.58·45-s − 2.70·47-s − 15.8·51-s + 1.64·53-s + 1.64·55-s − 14.9·57-s − 4.64·59-s + 14.2·61-s − 8.22·65-s + ⋯ |
| L(s) = 1 | − 1.52·3-s − 0.736·5-s + 1.33·9-s − 0.301·11-s + 1.38·13-s + 1.12·15-s + 1.45·17-s + 1.29·19-s − 0.343·23-s − 0.458·25-s − 0.509·27-s + 1.16·29-s + 0.718·31-s + 0.460·33-s + 0.599·37-s − 2.11·39-s + 1.70·41-s + 0.609·43-s − 0.981·45-s − 0.395·47-s − 2.22·51-s + 0.226·53-s + 0.221·55-s − 1.97·57-s − 0.604·59-s + 1.82·61-s − 1.02·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256892174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256892174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 3.64T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 + 4.64T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 4.35T + 71T^{2} \) |
| 73 | \( 1 - 0.354T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 + 2.70T + 83T^{2} \) |
| 89 | \( 1 - 6.58T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.904806215933294428776751238964, −6.96247438796743979447106023372, −6.25589967543361127439518887901, −5.69003880685543450766270264325, −5.20638565994174340742188263959, −4.28923402161824780917008581609, −3.68798666995465885207399270648, −2.77584002743352011984606403540, −1.18440293302880180007687105505, −0.73560815900512632775081729613,
0.73560815900512632775081729613, 1.18440293302880180007687105505, 2.77584002743352011984606403540, 3.68798666995465885207399270648, 4.28923402161824780917008581609, 5.20638565994174340742188263959, 5.69003880685543450766270264325, 6.25589967543361127439518887901, 6.96247438796743979447106023372, 7.904806215933294428776751238964
