L(s) = 1 | + 0.517i·2-s + 1.73·4-s + (−1 + 2.44i)7-s + 1.93i·8-s + 0.378i·11-s + 1.79i·13-s + (−1.26 − 0.517i)14-s + 2.46·16-s − 3.46·17-s + 1.79i·19-s − 0.196·22-s + 1.41i·23-s − 0.928·26-s + (−1.73 + 4.24i)28-s − 1.41i·29-s + ⋯ |
L(s) = 1 | + 0.366i·2-s + 0.866·4-s + (−0.377 + 0.925i)7-s + 0.683i·8-s + 0.114i·11-s + 0.497i·13-s + (−0.338 − 0.138i)14-s + 0.616·16-s − 0.840·17-s + 0.411i·19-s − 0.0418·22-s + 0.294i·23-s − 0.182·26-s + (−0.327 + 0.801i)28-s − 0.262i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.687922679\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.687922679\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
good | 2 | \( 1 - 0.517iT - 2T^{2} \) |
| 11 | \( 1 - 0.378iT - 11T^{2} \) |
| 13 | \( 1 - 1.79iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 1.79iT - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 6.69iT - 31T^{2} \) |
| 37 | \( 1 - 1.46T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 - 13.3iT - 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 15.0iT - 71T^{2} \) |
| 73 | \( 1 + 1.79iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 15.1iT - 97T^{2} \) |
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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
−9.694871595924946930839742592456, −8.700918763503966056084567345465, −8.208125394775006009215791099101, −7.01459636118908847318828430890, −6.61819148397085737027265794916, −5.72641893913960476278511251009, −4.97237481236319717081153515247, −3.63244661424104735421847718263, −2.59766341470026924219244363866, −1.73884013747167011942458183318,
0.60692318059298194346119915080, 1.97788079613858379426440910948, 3.03936652351935202388385817938, 3.85619590074714188386106387303, 4.90145846275400143737148333785, 6.14516664035955040737070642987, 6.71448721601969620544894681179, 7.47873852813941432752181179660, 8.242271791760575878122383453188, 9.381527166829061580367596824802