| L(s) = 1 | + (−2 − i)5-s + i·7-s − 11-s + i·13-s + 3i·17-s − 4·19-s − 2i·23-s + (3 + 4i)25-s − 29-s + 6·31-s + (1 − 2i)35-s + 2i·37-s + 10·41-s + 9i·47-s − 49-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s + 0.377i·7-s − 0.301·11-s + 0.277i·13-s + 0.727i·17-s − 0.917·19-s − 0.417i·23-s + (0.600 + 0.800i)25-s − 0.185·29-s + 1.07·31-s + (0.169 − 0.338i)35-s + 0.328i·37-s + 1.56·41-s + 1.31i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6161888241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6161888241\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 19iT - 97T^{2} \) |
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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
−7.982300696960332248178305040233, −7.51065617365675916708942451567, −6.43984449003284318264038877093, −5.97458484732942600484325972926, −4.80980032008825824645681724315, −4.42920133830405890444571571155, −3.51293061900044232850777348233, −2.60085696512579932956849030651, −1.52016929022024017850096624061, −0.19847007308049571313134729690,
0.965779983167552295787105672169, 2.44102233786440693658150136763, 3.10962613931106203746762076164, 4.10445279714158812480481183592, 4.56039797867431465394370343944, 5.62036477641083055526104219028, 6.37822251676539880714682177670, 7.26736350244204974636210448635, 7.57460813863053973245246217736, 8.395565788404518260586571605794
