| L(s) = 1 | + (−1.07 − 0.921i)2-s + (0.303 + 1.97i)4-s + i·5-s + (1.82 − 1.91i)7-s + (1.49 − 2.40i)8-s + (0.921 − 1.07i)10-s − 6.24i·11-s − 2.40i·13-s + (−3.72 + 0.373i)14-s + (−3.81 + 1.19i)16-s + 1.30i·17-s + 3.94·19-s + (−1.97 + 0.303i)20-s + (−5.74 + 6.69i)22-s + 3.55i·23-s + ⋯ |
| L(s) = 1 | + (−0.758 − 0.651i)2-s + (0.151 + 0.988i)4-s + 0.447i·5-s + (0.690 − 0.723i)7-s + (0.528 − 0.848i)8-s + (0.291 − 0.339i)10-s − 1.88i·11-s − 0.666i·13-s + (−0.995 + 0.0997i)14-s + (−0.954 + 0.299i)16-s + 0.317i·17-s + 0.905·19-s + (−0.442 + 0.0677i)20-s + (−1.22 + 1.42i)22-s + 0.741i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9869863869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9869863869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.07 + 0.921i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-1.82 + 1.91i)T \) |
| good | 11 | \( 1 + 6.24iT - 11T^{2} \) |
| 13 | \( 1 + 2.40iT - 13T^{2} \) |
| 17 | \( 1 - 1.30iT - 17T^{2} \) |
| 19 | \( 1 - 3.94T + 19T^{2} \) |
| 23 | \( 1 - 3.55iT - 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 + 1.26iT - 41T^{2} \) |
| 43 | \( 1 + 2.19iT - 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 0.415T + 59T^{2} \) |
| 61 | \( 1 + 12.6iT - 61T^{2} \) |
| 67 | \( 1 + 3.10iT - 67T^{2} \) |
| 71 | \( 1 + 10.4iT - 71T^{2} \) |
| 73 | \( 1 + 1.64iT - 73T^{2} \) |
| 79 | \( 1 + 9.18iT - 79T^{2} \) |
| 83 | \( 1 - 7.39T + 83T^{2} \) |
| 89 | \( 1 + 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−9.381206685658348352301697771949, −8.677210525629967619578701490213, −7.76786657938386392465521164768, −7.39468889050557026902291444415, −6.14194779357999031084192495117, −5.16399698828556954473257136145, −3.60475489383521346940886552595, −3.33672065753020288777987859850, −1.79024571694650763305685482426, −0.55370687474428030260207714205,
1.50765809878986385831296269855, 2.31159566732134880403207509600, 4.31705966037614199373596755581, 5.04887455282325269997835255730, 5.76049295867320916663030440787, 7.04183621995883233249140762691, 7.41354939706118479578494373375, 8.407989631386667806573274827584, 9.218014373088256773252077714762, 9.582278595009961120785281172720
