| L(s) = 1 | + i·2-s − 3-s − 4-s + 3.32i·5-s − i·6-s − i·8-s + 9-s − 3.32·10-s − 0.398i·11-s + 12-s + (−3.32 + 1.39i)13-s − 3.32i·15-s + 16-s − 2.39·17-s + i·18-s + 7.04i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 1.48i·5-s − 0.408i·6-s − 0.353i·8-s + 0.333·9-s − 1.05·10-s − 0.120i·11-s + 0.288·12-s + (−0.921 + 0.387i)13-s − 0.858i·15-s + 0.250·16-s − 0.581·17-s + 0.235i·18-s + 1.61i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1618844278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1618844278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (3.32 - 1.39i)T \) |
| good | 5 | \( 1 - 3.32iT - 5T^{2} \) |
| 11 | \( 1 + 0.398iT - 11T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 - 7.04iT - 19T^{2} \) |
| 23 | \( 1 + 0.676T + 23T^{2} \) |
| 29 | \( 1 - 4.11T + 29T^{2} \) |
| 31 | \( 1 + 2.64iT - 31T^{2} \) |
| 37 | \( 1 + 1.32iT - 37T^{2} \) |
| 41 | \( 1 + 0.646iT - 41T^{2} \) |
| 43 | \( 1 + 8.89T + 43T^{2} \) |
| 47 | \( 1 - 1.20iT - 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 59 | \( 1 - 4.64iT - 59T^{2} \) |
| 61 | \( 1 + 2.11T + 61T^{2} \) |
| 67 | \( 1 - 8.49iT - 67T^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 5.60iT - 73T^{2} \) |
| 79 | \( 1 - 4.79T + 79T^{2} \) |
| 83 | \( 1 - 9.44iT - 83T^{2} \) |
| 89 | \( 1 + 9.44iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.087027516539329781398520492203, −8.012388871998751400408700903194, −7.53285739466591262128728828521, −6.68305974301093102655219659979, −6.37682866924324116964643893186, −5.57766963906422587211961790759, −4.66011312692943668328688600483, −3.82121580016717315797552542784, −2.89765341169371013131703556653, −1.80039207785902550726345941873,
0.06027307661935353412901108695, 0.958374883123755334598654783219, 2.02257842126672150961273454223, 3.06645459041822135974328876064, 4.30810462239104586934626041806, 4.93100100631546539395595214837, 5.15372099228138256343241594866, 6.34538900819411581383986856372, 7.15875232841033681669292555823, 8.155668053616188584923420991835
