L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.0653 − 2.23i)5-s + (0.522 + 0.522i)7-s + 1.00i·9-s + 3.66i·11-s + (0.950 + 0.950i)13-s + (−1.62 + 1.53i)15-s + (3.43 + 3.43i)17-s + 1.65·19-s − 0.738i·21-s + (−0.329 + 4.78i)23-s + (−4.99 − 0.292i)25-s + (0.707 − 0.707i)27-s + 1.81i·29-s − 1.24·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.0292 − 0.999i)5-s + (0.197 + 0.197i)7-s + 0.333i·9-s + 1.10i·11-s + (0.263 + 0.263i)13-s + (−0.420 + 0.396i)15-s + (0.832 + 0.832i)17-s + 0.379·19-s − 0.161i·21-s + (−0.0687 + 0.997i)23-s + (−0.998 − 0.0584i)25-s + (0.136 − 0.136i)27-s + 0.336i·29-s − 0.223·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.455390390\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.455390390\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.0653 + 2.23i)T \) |
| 23 | \( 1 + (0.329 - 4.78i)T \) |
good | 7 | \( 1 + (-0.522 - 0.522i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.66iT - 11T^{2} \) |
| 13 | \( 1 + (-0.950 - 0.950i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.43 - 3.43i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 29 | \( 1 - 1.81iT - 29T^{2} \) |
| 31 | \( 1 + 1.24T + 31T^{2} \) |
| 37 | \( 1 + (-2.81 - 2.81i)T + 37iT^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + (7.63 - 7.63i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.98 + 6.98i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.34 + 1.34i)T - 53iT^{2} \) |
| 59 | \( 1 + 1.65iT - 59T^{2} \) |
| 61 | \( 1 + 2.65iT - 61T^{2} \) |
| 67 | \( 1 + (-5.90 - 5.90i)T + 67iT^{2} \) |
| 71 | \( 1 - 6.37T + 71T^{2} \) |
| 73 | \( 1 + (11.2 + 11.2i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.17T + 79T^{2} \) |
| 83 | \( 1 + (8.40 - 8.40i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.81T + 89T^{2} \) |
| 97 | \( 1 + (-0.249 - 0.249i)T + 97iT^{2} \) |
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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
−9.619630163907020575466130354183, −8.766798441009729825581499595737, −7.910409078047034781181338913463, −7.30223837892293439032974654290, −6.17508567498307573866862134059, −5.42211989327799725327993596059, −4.67258103552201573444076091938, −3.67289856263487649724787113690, −2.04320913397630800073107332257, −1.16392273028792000485103060582,
0.75327501872605367889609000957, 2.61391260354541415156999704718, 3.42746834907416097457785906620, 4.39358408753188677036563194274, 5.63192854660864734270471220525, 6.07292651605850441598118321770, 7.15054414862135029025393886454, 7.82534997225152299842474370475, 8.836959539754810406681953092359, 9.713735530431733440985257322725