| L(s) = 1 | + 5-s + 4.85i·7-s − 5.32·11-s − 0.812·13-s − 0.626·17-s + 6.95i·19-s + (−4.49 − 1.66i)23-s + 25-s − 2.93i·29-s + 5.30·31-s + 4.85i·35-s − 8.73i·37-s + 4.04i·41-s + 2.20i·43-s − 2.43i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.83i·7-s − 1.60·11-s − 0.225·13-s − 0.152·17-s + 1.59i·19-s + (−0.937 − 0.347i)23-s + 0.200·25-s − 0.544i·29-s + 0.952·31-s + 0.821i·35-s − 1.43i·37-s + 0.631i·41-s + 0.335i·43-s − 0.355i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2622445899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2622445899\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + (4.49 + 1.66i)T \) |
| good | 7 | \( 1 - 4.85iT - 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 + 0.812T + 13T^{2} \) |
| 17 | \( 1 + 0.626T + 17T^{2} \) |
| 19 | \( 1 - 6.95iT - 19T^{2} \) |
| 29 | \( 1 + 2.93iT - 29T^{2} \) |
| 31 | \( 1 - 5.30T + 31T^{2} \) |
| 37 | \( 1 + 8.73iT - 37T^{2} \) |
| 41 | \( 1 - 4.04iT - 41T^{2} \) |
| 43 | \( 1 - 2.20iT - 43T^{2} \) |
| 47 | \( 1 + 2.43iT - 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 59 | \( 1 + 4.20iT - 59T^{2} \) |
| 61 | \( 1 + 12.6iT - 61T^{2} \) |
| 67 | \( 1 - 11.8iT - 67T^{2} \) |
| 71 | \( 1 + 5.69iT - 71T^{2} \) |
| 73 | \( 1 + 7.08T + 73T^{2} \) |
| 79 | \( 1 + 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 2.07T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 3.75iT - 97T^{2} \) |
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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
−8.711369435548700798114300898366, −8.164769365197978625853425382006, −7.65701302456643660368377122967, −6.34778201640265205359957933418, −5.84239215892803164602994768525, −5.34713091066577382726557394709, −4.51451231567257778430330499943, −3.22610827288618250099429508697, −2.41859258470744818323624271082, −1.90064627761230131758709384547,
0.07276315323142969719549043454, 1.17614412861577891916605457945, 2.46250139947397088219587005734, 3.22414955636484234541847496198, 4.36461819177089847452406667439, 4.82761457288954621436368709752, 5.71341528313868237632856670011, 6.74703927240696543802507304989, 7.20561227276236454011586959867, 7.896926279936373374948094068106
