| L(s) = 1 | − i·3-s + 5-s + (−1.73 + 2i)7-s + 2·9-s − 5.19·11-s + 13-s − i·15-s − 1.73i·17-s − 2i·19-s + (2 + 1.73i)21-s + 25-s − 5i·27-s − 1.73i·29-s + 3.46·31-s + 5.19i·33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447·5-s + (−0.654 + 0.755i)7-s + 0.666·9-s − 1.56·11-s + 0.277·13-s − 0.258i·15-s − 0.420i·17-s − 0.458i·19-s + (0.436 + 0.377i)21-s + 0.200·25-s − 0.962i·27-s − 0.321i·29-s + 0.622·31-s + 0.904i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0716 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0716 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.452828859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.452828859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 1.73iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 - iT - 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−8.841000157553617884602501693096, −8.039913567405794342920301803144, −7.23381743672276428264402342361, −6.59785326360564507107211078629, −5.65489848596982778514886741454, −5.13266520978742060342733363730, −3.88791233560929109298455440758, −2.64757529371516388409776438228, −2.14200426539424490131240110686, −0.53844007915995601454196239166,
1.18744683661151100937232727541, 2.59378659542787943891821898824, 3.50877788849952376035444208309, 4.40079957013832487509431369616, 5.16489170313982877004621355497, 6.07793787880874468408217800109, 6.88141901062599380008203926700, 7.70162974795552409214636140933, 8.418557479524906328547120029206, 9.472526967871502125370385007777
