| L(s) = 1 | − 1.32i·3-s + i·5-s − 3.37·7-s + 1.23·9-s + 1.86·11-s − 4.25·13-s + 1.32·15-s + 3.38i·17-s + 4.52·19-s + 4.48i·21-s + (3.47 − 3.30i)23-s − 25-s − 5.62i·27-s + 3.15·29-s − 2.04i·31-s + ⋯ |
| L(s) = 1 | − 0.767i·3-s + 0.447i·5-s − 1.27·7-s + 0.411·9-s + 0.563·11-s − 1.17·13-s + 0.343·15-s + 0.821i·17-s + 1.03·19-s + 0.978i·21-s + (0.725 − 0.688i)23-s − 0.200·25-s − 1.08i·27-s + 0.585·29-s − 0.367i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.503406441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503406441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-3.47 + 3.30i)T \) |
| good | 3 | \( 1 + 1.32iT - 3T^{2} \) |
| 7 | \( 1 + 3.37T + 7T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 - 3.38iT - 17T^{2} \) |
| 19 | \( 1 - 4.52T + 19T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 31 | \( 1 + 2.04iT - 31T^{2} \) |
| 37 | \( 1 - 5.40iT - 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 - 2.07T + 43T^{2} \) |
| 47 | \( 1 + 5.06iT - 47T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 + 11.1iT - 59T^{2} \) |
| 61 | \( 1 - 4.01iT - 61T^{2} \) |
| 67 | \( 1 - 5.68T + 67T^{2} \) |
| 71 | \( 1 - 1.69iT - 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 - 4.32iT - 89T^{2} \) |
| 97 | \( 1 - 8.95iT - 97T^{2} \) |
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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
−9.374034487824078928130986538936, −8.190430010733782781985006355758, −7.40821954511336955740502216351, −6.67243721522829603832257079241, −6.38023852593951602279808858576, −5.16651295350270188253664172776, −4.02952270219281773979966307038, −3.08365226954376290819779885649, −2.16286175633634530613644135248, −0.74250232209659705950779014255,
0.949643055646868203443891474862, 2.66889363989104940296149110249, 3.50712978963432914048657722243, 4.44555166490867254624717196294, 5.15307441392370737110294399586, 6.07732064234619970283956098694, 7.15879830875309780675122337999, 7.52225696260751754844620985853, 9.104064405996316975975563035588, 9.346807338610881487993654974680
