| L(s) = 1 | − 2-s + 4-s − 5-s − 0.0653i·7-s − 8-s + 10-s + 1.77·11-s − 2.43i·13-s + 0.0653i·14-s + 16-s − 0.153i·17-s − 7.53·19-s − 20-s − 1.77·22-s + 2.47i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.0246i·7-s − 0.353·8-s + 0.316·10-s + 0.534·11-s − 0.674i·13-s + 0.0174i·14-s + 0.250·16-s − 0.0373i·17-s − 1.72·19-s − 0.223·20-s − 0.378·22-s + 0.515i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019149148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019149148\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 67 | \( 1 + (1.89 + 7.96i)T \) |
| good | 7 | \( 1 + 0.0653iT - 7T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 2.43iT - 13T^{2} \) |
| 17 | \( 1 + 0.153iT - 17T^{2} \) |
| 19 | \( 1 + 7.53T + 19T^{2} \) |
| 23 | \( 1 - 2.47iT - 23T^{2} \) |
| 29 | \( 1 - 4.70iT - 29T^{2} \) |
| 31 | \( 1 + 2.93iT - 31T^{2} \) |
| 37 | \( 1 + 7.68T + 37T^{2} \) |
| 41 | \( 1 - 6.63T + 41T^{2} \) |
| 43 | \( 1 - 4.71iT - 43T^{2} \) |
| 47 | \( 1 - 6.51iT - 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 0.249iT - 59T^{2} \) |
| 61 | \( 1 + 4.77iT - 61T^{2} \) |
| 71 | \( 1 + 0.869iT - 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 + 6.75iT - 79T^{2} \) |
| 83 | \( 1 + 0.191iT - 83T^{2} \) |
| 89 | \( 1 + 9.07iT - 89T^{2} \) |
| 97 | \( 1 - 5.89iT - 97T^{2} \) |
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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
−7.996298831033143895206659615129, −7.50810698790463336132308723513, −6.69216279555396594676828558662, −6.11219899846012673826625295610, −5.23057091417069253069976316060, −4.27143022979845435558936078175, −3.56460016448541163434837242459, −2.61692976614253214853964047135, −1.65839101982301291867951641116, −0.53069035743294105671884000459,
0.64953858278157541251440367879, 1.88577403243008203312241386085, 2.57681974766566256422770516099, 3.87164964511483115933828986647, 4.21911401339747240823742971078, 5.34631472527509111569368651557, 6.23504160372768052158804112073, 6.84724579426637238481657922285, 7.34970886730048476724848655420, 8.419908813297948127317855856932
