| L(s) = 1 | + 2.43·2-s + 3-s + 3.90·4-s + 1.43·5-s + 2.43·6-s − 7-s + 4.64·8-s + 9-s + 3.47·10-s + 3.90·12-s + 2.30·13-s − 2.43·14-s + 1.43·15-s + 3.46·16-s + 1.14·17-s + 2.43·18-s + 5.47·19-s + 5.59·20-s − 21-s + 3.00·23-s + 4.64·24-s − 2.95·25-s + 5.59·26-s + 27-s − 3.90·28-s − 3.16·29-s + 3.47·30-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.95·4-s + 0.639·5-s + 0.992·6-s − 0.377·7-s + 1.64·8-s + 0.333·9-s + 1.09·10-s + 1.12·12-s + 0.638·13-s − 0.649·14-s + 0.369·15-s + 0.866·16-s + 0.277·17-s + 0.572·18-s + 1.25·19-s + 1.25·20-s − 0.218·21-s + 0.626·23-s + 0.947·24-s − 0.590·25-s + 1.09·26-s + 0.192·27-s − 0.738·28-s − 0.588·29-s + 0.635·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.995620498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.995620498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.43T + 2T^{2} \) |
| 5 | \( 1 - 1.43T + 5T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 23 | \( 1 - 3.00T + 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 + 6.99T + 31T^{2} \) |
| 37 | \( 1 + 8.16T + 37T^{2} \) |
| 41 | \( 1 - 5.59T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 9.28T + 53T^{2} \) |
| 59 | \( 1 + 6.89T + 59T^{2} \) |
| 61 | \( 1 + 8.50T + 61T^{2} \) |
| 67 | \( 1 - 7.61T + 67T^{2} \) |
| 71 | \( 1 + 1.92T + 71T^{2} \) |
| 73 | \( 1 - 4.83T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 9.40T + 83T^{2} \) |
| 89 | \( 1 + 4.35T + 89T^{2} \) |
| 97 | \( 1 + 6.56T + 97T^{2} \) |
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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
−9.109190844131529038629068857786, −7.83485887884616410472551036044, −7.15330422243976034997586780478, −6.34606124908352852091214131859, −5.60519099053935560194585314743, −5.07591951749177202381743219898, −3.88282353355599189273972139150, −3.41234345253578277368079241114, −2.52309959858540580708823373241, −1.53269902999755525942341424192,
1.53269902999755525942341424192, 2.52309959858540580708823373241, 3.41234345253578277368079241114, 3.88282353355599189273972139150, 5.07591951749177202381743219898, 5.60519099053935560194585314743, 6.34606124908352852091214131859, 7.15330422243976034997586780478, 7.83485887884616410472551036044, 9.109190844131529038629068857786
