| L(s) = 1 | + 1.36·2-s − 0.149·4-s + 3.81·5-s + 4.20·7-s − 2.92·8-s + 5.18·10-s + 5.13·11-s − 2.63·13-s + 5.71·14-s − 3.67·16-s − 5.08·17-s − 3.48·19-s − 0.571·20-s + 6.97·22-s + 6.87·23-s + 9.52·25-s − 3.58·26-s − 0.629·28-s − 8.87·29-s + 2.71·31-s + 0.846·32-s − 6.92·34-s + 16.0·35-s − 6.12·37-s − 4.73·38-s − 11.1·40-s + 2.10·41-s + ⋯ |
| L(s) = 1 | + 0.961·2-s − 0.0749·4-s + 1.70·5-s + 1.58·7-s − 1.03·8-s + 1.63·10-s + 1.54·11-s − 0.731·13-s + 1.52·14-s − 0.919·16-s − 1.23·17-s − 0.798·19-s − 0.127·20-s + 1.48·22-s + 1.43·23-s + 1.90·25-s − 0.703·26-s − 0.119·28-s − 1.64·29-s + 0.488·31-s + 0.149·32-s − 1.18·34-s + 2.70·35-s − 1.00·37-s − 0.768·38-s − 1.76·40-s + 0.329·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.593721351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.593721351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 7 | \( 1 - 4.20T + 7T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 + 2.63T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 3.48T + 19T^{2} \) |
| 23 | \( 1 - 6.87T + 23T^{2} \) |
| 29 | \( 1 + 8.87T + 29T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + 6.12T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 + 9.74T + 43T^{2} \) |
| 47 | \( 1 + 3.39T + 47T^{2} \) |
| 53 | \( 1 + 3.88T + 53T^{2} \) |
| 59 | \( 1 + 0.860T + 59T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 3.13T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 - 4.56T + 97T^{2} \) |
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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
−9.571143945369206604697479384145, −9.116508065865961296947016255111, −8.417086120822406996353664010428, −6.86892279033189515683535416262, −6.34843738282850376438975579419, −5.18913466499037612302462672663, −4.92033919684210070008771694236, −3.86876024447213695740776274486, −2.36325265638016774757450077140, −1.56916508904111817377324564889,
1.56916508904111817377324564889, 2.36325265638016774757450077140, 3.86876024447213695740776274486, 4.92033919684210070008771694236, 5.18913466499037612302462672663, 6.34843738282850376438975579419, 6.86892279033189515683535416262, 8.417086120822406996353664010428, 9.116508065865961296947016255111, 9.571143945369206604697479384145
