| L(s) = 1 | + 0.690·2-s − 0.694·3-s − 1.52·4-s + 5-s − 0.479·6-s − 2.33·7-s − 2.43·8-s − 2.51·9-s + 0.690·10-s + 4.57·11-s + 1.05·12-s + 1.28·13-s − 1.61·14-s − 0.694·15-s + 1.36·16-s − 0.654·17-s − 1.73·18-s − 1.52·20-s + 1.62·21-s + 3.15·22-s − 5.56·23-s + 1.68·24-s + 25-s + 0.884·26-s + 3.83·27-s + 3.55·28-s − 4.73·29-s + ⋯ |
| L(s) = 1 | + 0.487·2-s − 0.401·3-s − 0.761·4-s + 0.447·5-s − 0.195·6-s − 0.882·7-s − 0.859·8-s − 0.839·9-s + 0.218·10-s + 1.38·11-s + 0.305·12-s + 0.355·13-s − 0.430·14-s − 0.179·15-s + 0.342·16-s − 0.158·17-s − 0.409·18-s − 0.340·20-s + 0.353·21-s + 0.673·22-s − 1.16·23-s + 0.344·24-s + 0.200·25-s + 0.173·26-s + 0.737·27-s + 0.672·28-s − 0.878·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.292404102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.292404102\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.690T + 2T^{2} \) |
| 3 | \( 1 + 0.694T + 3T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 + 0.654T + 17T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 4.48T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 - 6.60T + 53T^{2} \) |
| 59 | \( 1 - 5.89T + 59T^{2} \) |
| 61 | \( 1 + 7.49T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 3.75T + 71T^{2} \) |
| 73 | \( 1 - 7.96T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 9.83T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 4.20T + 97T^{2} \) |
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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
−9.255286686215132646152601231862, −8.802085300841517149863308695478, −7.68325791891429897736156379653, −6.41245865611690843485846763912, −6.04998521903949911720497515486, −5.38962822558292841772405043940, −4.14368982210964638394511360889, −3.65060998325161249492532421557, −2.44228204446614509098300381439, −0.73051962673177732544841938186,
0.73051962673177732544841938186, 2.44228204446614509098300381439, 3.65060998325161249492532421557, 4.14368982210964638394511360889, 5.38962822558292841772405043940, 6.04998521903949911720497515486, 6.41245865611690843485846763912, 7.68325791891429897736156379653, 8.802085300841517149863308695478, 9.255286686215132646152601231862
