| L(s) = 1 | + 2-s + 4-s + (−1.32 − 1.79i)5-s + 1.87i·7-s + 8-s + (−1.32 − 1.79i)10-s + 1.51·11-s − 4.78·13-s + 1.87i·14-s + 16-s + 3.99·17-s − 4.68i·19-s + (−1.32 − 1.79i)20-s + 1.51·22-s + 6.51·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.594 − 0.804i)5-s + 0.707i·7-s + 0.353·8-s + (−0.420 − 0.568i)10-s + 0.455·11-s − 1.32·13-s + 0.500i·14-s + 0.250·16-s + 0.968·17-s − 1.07i·19-s + (−0.297 − 0.402i)20-s + 0.322·22-s + 1.35·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.444932590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.444932590\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.32 + 1.79i)T \) |
| 37 | \( 1 + (6.03 + 0.734i)T \) |
| good | 7 | \( 1 - 1.87iT - 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 + 4.78T + 13T^{2} \) |
| 17 | \( 1 - 3.99T + 17T^{2} \) |
| 19 | \( 1 + 4.68iT - 19T^{2} \) |
| 23 | \( 1 - 6.51T + 23T^{2} \) |
| 29 | \( 1 - 0.200iT - 29T^{2} \) |
| 31 | \( 1 + 5.79iT - 31T^{2} \) |
| 41 | \( 1 + 4.10T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 5.15iT - 47T^{2} \) |
| 53 | \( 1 - 2.22iT - 53T^{2} \) |
| 59 | \( 1 + 13.2iT - 59T^{2} \) |
| 61 | \( 1 + 4.09iT - 61T^{2} \) |
| 67 | \( 1 - 5.76iT - 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 + 14.2iT - 73T^{2} \) |
| 79 | \( 1 - 5.95iT - 79T^{2} \) |
| 83 | \( 1 + 9.42iT - 83T^{2} \) |
| 89 | \( 1 + 5.47iT - 89T^{2} \) |
| 97 | \( 1 + 2.03T + 97T^{2} \) |
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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
−8.547602993913687112303293048084, −7.54405817729893735446215015681, −7.14128021518322835524803436213, −6.08616880899787801744027366793, −5.10368744178850519395085407866, −4.93444687345582965377602285145, −3.86273068684241691271527418487, −2.99007416605756550884116485121, −2.03712547162275820686665793848, −0.64887995651492866404113936017,
1.14583191193794361564757914542, 2.52859904882223674554327971427, 3.34518976526533772017203946404, 3.98513984547055612941260506109, 4.84894413919474982354790519948, 5.66173945148372889137419942258, 6.64353501225479396494984560712, 7.27736565578626941990260509173, 7.60001823854311689640549253310, 8.616721513367551264788163184193
