| L(s) = 1 | + 0.223i·2-s + 1.94·4-s + 3.02·5-s + 0.884i·8-s + 0.677i·10-s + 2.32i·11-s + 6.38i·13-s + 3.70·16-s + 1.14·17-s − 3.16i·19-s + 5.89·20-s − 0.520·22-s + 4.24i·23-s + 4.15·25-s − 1.43·26-s + ⋯ |
| L(s) = 1 | + 0.158i·2-s + 0.974·4-s + 1.35·5-s + 0.312i·8-s + 0.214i·10-s + 0.700i·11-s + 1.77i·13-s + 0.925·16-s + 0.278·17-s − 0.727i·19-s + 1.31·20-s − 0.110·22-s + 0.885i·23-s + 0.830·25-s − 0.280·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.192568283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.192568283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.223iT - 2T^{2} \) |
| 5 | \( 1 - 3.02T + 5T^{2} \) |
| 11 | \( 1 - 2.32iT - 11T^{2} \) |
| 13 | \( 1 - 6.38iT - 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 + 3.16iT - 19T^{2} \) |
| 23 | \( 1 - 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 2.88iT - 29T^{2} \) |
| 31 | \( 1 + 5.49iT - 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 + 2.57T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 10.8iT - 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 7.99iT - 61T^{2} \) |
| 67 | \( 1 - 1.22T + 67T^{2} \) |
| 71 | \( 1 + 9.56iT - 71T^{2} \) |
| 73 | \( 1 + 10.2iT - 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−9.127895784805767307731725818785, −7.82724370992712053796331790349, −7.24690881890973293620644073403, −6.35165504415912638840511751448, −6.12038386054480304551855787229, −5.08444234641394028950012973264, −4.24148142950178159152783276586, −2.92139401412185638312102769105, −2.05185476366981565710495372493, −1.54526415411717068051968394079,
0.965201064261632322533698494731, 1.93583459807918275967722989088, 2.90154198033088048894670435462, 3.42881779300793360877832794096, 5.02907890168126086958221153371, 5.69599298625395623607872714104, 6.20911157062682534169362406784, 6.90963231485457252673752039102, 8.001205366762866340220161334975, 8.387873561811956535451039819212
