| L(s) = 1 | + (1.16 − 2.37i)7-s − 3.74i·11-s + 0.841i·13-s − 3.36·17-s + 4.55i·19-s + 7.64i·23-s + 1.41i·29-s + 0.979i·31-s + 2.32·37-s − 10.3·41-s − 10.8·43-s − 7.91·47-s + (−4.29 − 5.53i)49-s − 4.35i·53-s − 1.38·59-s + ⋯ |
| L(s) = 1 | + (0.439 − 0.898i)7-s − 1.12i·11-s + 0.233i·13-s − 0.814·17-s + 1.04i·19-s + 1.59i·23-s + 0.262i·29-s + 0.175i·31-s + 0.382·37-s − 1.61·41-s − 1.64·43-s − 1.15·47-s + (−0.613 − 0.790i)49-s − 0.598i·53-s − 0.180·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6175936900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6175936900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.16 + 2.37i)T \) |
| good | 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 0.841iT - 13T^{2} \) |
| 17 | \( 1 + 3.36T + 17T^{2} \) |
| 19 | \( 1 - 4.55iT - 19T^{2} \) |
| 23 | \( 1 - 7.64iT - 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 0.979iT - 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 7.91T + 47T^{2} \) |
| 53 | \( 1 + 4.35iT - 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 - 8.66iT - 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 - 14.8iT - 97T^{2} \) |
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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
−8.318651018192323694390517125456, −7.60372928477890423303528938910, −6.85514338469580961914029978686, −6.23806192992237881097600107875, −5.34580066561711218479200812500, −4.74224394663776030668459058424, −3.63335391217431271200863470339, −3.41825234689233379352491753210, −1.96012887075746806765886290562, −1.19104905627994129194414164335,
0.15143630591804650196217524484, 1.72441747775670133686493501097, 2.37684751354974617184447420383, 3.18419626961636177531540168796, 4.57379344546439135919402297533, 4.68198903634134889193480815862, 5.61022657198672047771575877820, 6.59777208857896363499118924013, 6.89763799900865535990490302079, 7.941397059243144738432095291105
