L(s) = 1 | + (−1.08 + 1.95i)5-s + (0.595 + 2.57i)7-s − 3.74i·11-s + 3.36·13-s + 0.841i·17-s + 5.59i·19-s + 2.35·23-s + (−2.64 − 4.24i)25-s + 1.41i·29-s + 8.66i·31-s + (−5.68 − 1.63i)35-s + 5.15i·37-s − 5.74·41-s − 3.32i·43-s + 6.43i·47-s + ⋯ |
L(s) = 1 | + (−0.485 + 0.874i)5-s + (0.224 + 0.974i)7-s − 1.12i·11-s + 0.931·13-s + 0.204i·17-s + 1.28i·19-s + 0.490·23-s + (−0.529 − 0.848i)25-s + 0.262i·29-s + 1.55i·31-s + (−0.961 − 0.276i)35-s + 0.847i·37-s − 0.896·41-s − 0.507i·43-s + 0.938i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.302755671\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.302755671\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.08 - 1.95i)T \) |
| 7 | \( 1 + (-0.595 - 2.57i)T \) |
good | 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 - 0.841iT - 17T^{2} \) |
| 19 | \( 1 - 5.59iT - 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 5.15iT - 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 + 3.32iT - 43T^{2} \) |
| 47 | \( 1 - 6.43iT - 47T^{2} \) |
| 53 | \( 1 + 9.64T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 1.82iT - 67T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 + 0.979T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 - 2.16T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−10.08050190602219012373547331588, −8.856186329587849011837341780729, −8.409953946949810770640104335311, −7.60892319807949756348073304889, −6.39659230955118064599436539417, −6.00206065037844486883715751220, −4.89798253665673100177627838360, −3.52049407357390146940609419564, −3.05530346631825782266923445239, −1.55836388771196023868429407682,
0.57375556600070355221781668195, 1.83012044636770543257035936083, 3.43219932533775573895204928714, 4.43668604052307025623923310844, 4.85662378636721193185721717583, 6.14515705789965360824925371513, 7.21659122036695985144050566513, 7.68991939336680512431756938813, 8.676019083815234105086207858266, 9.373579260619006376153230925725