| L(s) = 1 | − 3.55i·5-s + 3.63i·7-s + 4.12·11-s − 1.20·13-s − 1.58i·17-s + i·19-s + 3.68·23-s − 7.67·25-s − 0.561i·29-s − 5.20i·31-s + 12.9·35-s + 9.27·37-s − 3.72i·41-s + 9.46i·43-s + 6.33·47-s + ⋯ |
| L(s) = 1 | − 1.59i·5-s + 1.37i·7-s + 1.24·11-s − 0.334·13-s − 0.384i·17-s + 0.229i·19-s + 0.767·23-s − 1.53·25-s − 0.104i·29-s − 0.934i·31-s + 2.18·35-s + 1.52·37-s − 0.582i·41-s + 1.44i·43-s + 0.924·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.957426715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.957426715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 + 3.55iT - 5T^{2} \) |
| 7 | \( 1 - 3.63iT - 7T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 + 1.58iT - 17T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 0.561iT - 29T^{2} \) |
| 31 | \( 1 + 5.20iT - 31T^{2} \) |
| 37 | \( 1 - 9.27T + 37T^{2} \) |
| 41 | \( 1 + 3.72iT - 41T^{2} \) |
| 43 | \( 1 - 9.46iT - 43T^{2} \) |
| 47 | \( 1 - 6.33T + 47T^{2} \) |
| 53 | \( 1 - 11.9iT - 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.60T + 61T^{2} \) |
| 67 | \( 1 - 2.06iT - 67T^{2} \) |
| 71 | \( 1 - 7.36T + 71T^{2} \) |
| 73 | \( 1 + 0.363T + 73T^{2} \) |
| 79 | \( 1 + 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 3.68T + 83T^{2} \) |
| 89 | \( 1 + 6.79iT - 89T^{2} \) |
| 97 | \( 1 - 8.86T + 97T^{2} \) |
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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
−9.064212386916457799186887633938, −8.178899285752152017138015368045, −7.38279294735929641177553898292, −6.12612358445930105806890545837, −5.74755077466702848550310129957, −4.75347897425140613147653356758, −4.27247413945371244155519863826, −2.94500889472223174135908120168, −1.86688738267607260649461388316, −0.836740506879374561629083131231,
0.983495118200961749660523739667, 2.30675557477197933864423597087, 3.41731812117891716464054121633, 3.86889094820802747090178702379, 4.87015881775769260836882502574, 6.17434035598865087918271788586, 6.76987768721924416453272901665, 7.17053533678756200238973266051, 7.902052754002358196731893814157, 8.996284168462172034187729728270
