| L(s) = 1 | − 1.23i·3-s + (−1 + 2i)5-s − 4.47i·7-s + 1.47·9-s + 3.23i·13-s + (2.47 + 1.23i)15-s − 6.47i·17-s + 19-s − 5.52·21-s + 2i·23-s + (−3 − 4i)25-s − 5.52i·27-s − 2·29-s + 1.52·31-s + (8.94 + 4.47i)35-s + ⋯ |
| L(s) = 1 | − 0.713i·3-s + (−0.447 + 0.894i)5-s − 1.69i·7-s + 0.490·9-s + 0.897i·13-s + (0.638 + 0.319i)15-s − 1.56i·17-s + 0.229·19-s − 1.20·21-s + 0.417i·23-s + (−0.600 − 0.800i)25-s − 1.06i·27-s − 0.371·29-s + 0.274·31-s + (1.51 + 0.755i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.310355790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.310355790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23iT - 3T^{2} \) |
| 7 | \( 1 + 4.47iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 4.76iT - 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 + 0.472iT - 43T^{2} \) |
| 47 | \( 1 + 12.4iT - 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 1.23iT - 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 + 6.47iT - 73T^{2} \) |
| 79 | \( 1 + 6.47T + 79T^{2} \) |
| 83 | \( 1 + 14.9iT - 83T^{2} \) |
| 89 | \( 1 - 6.94T + 89T^{2} \) |
| 97 | \( 1 - 4.18iT - 97T^{2} \) |
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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
−9.381271809379800535096420637953, −8.022533957142844517752540413513, −7.34401948694883130050108376551, −7.02333958766918565771175996025, −6.38917974441952988610920276054, −4.83756397347429447969902814539, −4.04581059692740183685151766327, −3.17061122152241894742194018719, −1.82542113696614157635501596830, −0.54003496349931498685870966663,
1.47694462615177501787957281178, 2.80689279243106968751623605986, 3.92019161124086504698463518031, 4.70923249489100859106265713991, 5.58091973879745802656206820661, 6.11796060274431820961882639778, 7.62497602293400837024737206692, 8.266233368501191408891085050228, 9.037983710308450042336330814107, 9.477123143720395338325765726226
