| L(s) = 1 | − 2·2-s − 5.10·3-s + 4·4-s − 18.4·5-s + 10.2·6-s + 1.23·7-s − 8·8-s − 0.894·9-s + 36.8·10-s + 71.5·11-s − 20.4·12-s + 63.7·13-s − 2.47·14-s + 94.0·15-s + 16·16-s + 2.24·17-s + 1.78·18-s + 89.0·19-s − 73.6·20-s − 6.31·21-s − 143.·22-s + 48.7·23-s + 40.8·24-s + 213.·25-s − 127.·26-s + 142.·27-s + 4.94·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.983·3-s + 0.5·4-s − 1.64·5-s + 0.695·6-s + 0.0667·7-s − 0.353·8-s − 0.0331·9-s + 1.16·10-s + 1.96·11-s − 0.491·12-s + 1.35·13-s − 0.0472·14-s + 1.61·15-s + 0.250·16-s + 0.0320·17-s + 0.0234·18-s + 1.07·19-s − 0.823·20-s − 0.0656·21-s − 1.38·22-s + 0.442·23-s + 0.347·24-s + 1.71·25-s − 0.961·26-s + 1.01·27-s + 0.0333·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1922 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.108236399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108236399\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 5.10T + 27T^{2} \) |
| 5 | \( 1 + 18.4T + 125T^{2} \) |
| 7 | \( 1 - 1.23T + 343T^{2} \) |
| 11 | \( 1 - 71.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.24T + 4.91e3T^{2} \) |
| 19 | \( 1 - 89.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 332.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 121.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 527.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 75.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 203.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 361.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 602.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 192.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 226.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 274.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 453.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 881.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 132.T + 9.12e5T^{2} \) |
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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
−8.808741812844539627883575855457, −8.105244133875803657200735780424, −7.30455620322055253090049053838, −6.48888487443582204610201844545, −5.97825546542706969547670444682, −4.65722103474084516230658295696, −3.89736506355091640568834025563, −3.08026228985463753627108024265, −1.08178930890848851844597566584, −0.77187764108356126076240130817,
0.77187764108356126076240130817, 1.08178930890848851844597566584, 3.08026228985463753627108024265, 3.89736506355091640568834025563, 4.65722103474084516230658295696, 5.97825546542706969547670444682, 6.48888487443582204610201844545, 7.30455620322055253090049053838, 8.105244133875803657200735780424, 8.808741812844539627883575855457
