L(s) = 1 | + (−0.5 + 0.866i)2-s + (2.99 + 0.803i)3-s + (−0.499 − 0.866i)4-s + (−1.15 + 1.91i)5-s + (−2.19 + 2.19i)6-s + (3.30 + 0.886i)7-s + 0.999·8-s + (5.75 + 3.32i)9-s + (−1.08 − 1.95i)10-s − 4.03i·11-s + (−0.803 − 2.99i)12-s + (−1.90 − 3.29i)13-s + (−2.42 + 2.42i)14-s + (−4.99 + 4.82i)15-s + (−0.5 + 0.866i)16-s + (−2.47 − 1.42i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (1.73 + 0.464i)3-s + (−0.249 − 0.433i)4-s + (−0.515 + 0.856i)5-s + (−0.896 + 0.896i)6-s + (1.25 + 0.335i)7-s + 0.353·8-s + (1.91 + 1.10i)9-s + (−0.342 − 0.618i)10-s − 1.21i·11-s + (−0.232 − 0.865i)12-s + (−0.527 − 0.914i)13-s + (−0.647 + 0.647i)14-s + (−1.29 + 1.24i)15-s + (−0.125 + 0.216i)16-s + (−0.600 − 0.346i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45371 + 1.24751i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45371 + 1.24751i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (1.15 - 1.91i)T \) |
| 37 | \( 1 + (-2.54 + 5.52i)T \) |
good | 3 | \( 1 + (-2.99 - 0.803i)T + (2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (-3.30 - 0.886i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + 4.03iT - 11T^{2} \) |
| 13 | \( 1 + (1.90 + 3.29i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.47 + 1.42i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.23 - 4.61i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + 5.69T + 23T^{2} \) |
| 29 | \( 1 + (2.30 - 2.30i)T - 29iT^{2} \) |
| 31 | \( 1 + (-1.95 - 1.95i)T + 31iT^{2} \) |
| 41 | \( 1 + (-4.19 + 2.42i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 8.10T + 43T^{2} \) |
| 47 | \( 1 + (5.47 - 5.47i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.82 - 1.82i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.59 + 1.23i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.89 + 10.8i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-1.41 + 5.28i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (5.05 + 8.75i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.11 - 7.11i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.88 - 10.7i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (-2.76 + 0.739i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (3.14 + 11.7i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 10.0iT - 97T^{2} \) |
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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
−11.23365031326221409416752141865, −10.52855572479519736942007916482, −9.547294682563532074853542581987, −8.478280537389265854083307633516, −8.039463125656311662219494123888, −7.45252148542844342634667682382, −5.88782678150151591807356641976, −4.46884345599656663618596867267, −3.38033248381223332786897627913, −2.19653038485025742509663271360,
1.58815060084271872489174256408, 2.37239615356342471780728398166, 4.23467006031677594109168411933, 4.46933952358407045033000730743, 7.07093166991569334450746195667, 7.78673083257901996472698978969, 8.421930107756002545661240056547, 9.209777252843165241379897927557, 9.924551003888456721343369588417, 11.37235839355424489154564326584