L(s) = 1 | − 2.60i·2-s − i·3-s − 4.77·4-s + i·5-s − 2.60·6-s + (2.60 − 0.474i)7-s + 7.22i·8-s − 9-s + 2.60·10-s + (3.23 − 0.726i)11-s + 4.77i·12-s + 6.07·13-s + (−1.23 − 6.77i)14-s + 15-s + 9.24·16-s + 4.27·17-s + ⋯ |
L(s) = 1 | − 1.84i·2-s − 0.577i·3-s − 2.38·4-s + 0.447i·5-s − 1.06·6-s + (0.983 − 0.179i)7-s + 2.55i·8-s − 0.333·9-s + 0.823·10-s + (0.975 − 0.219i)11-s + 1.37i·12-s + 1.68·13-s + (−0.330 − 1.81i)14-s + 0.258·15-s + 2.31·16-s + 1.03·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.790543720\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.790543720\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.60 + 0.474i)T \) |
| 11 | \( 1 + (-3.23 + 0.726i)T \) |
good | 2 | \( 1 + 2.60iT - 2T^{2} \) |
| 13 | \( 1 - 6.07T + 13T^{2} \) |
| 17 | \( 1 - 4.27T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 5.77T + 23T^{2} \) |
| 29 | \( 1 - 6.49iT - 29T^{2} \) |
| 31 | \( 1 + 2.66iT - 31T^{2} \) |
| 37 | \( 1 - 2.66T + 37T^{2} \) |
| 41 | \( 1 + 8.37T + 41T^{2} \) |
| 43 | \( 1 - 11.9iT - 43T^{2} \) |
| 47 | \( 1 + 12.6iT - 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 4.89iT - 59T^{2} \) |
| 61 | \( 1 - 7.86T + 61T^{2} \) |
| 67 | \( 1 - 2.92T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 4.53T + 73T^{2} \) |
| 79 | \( 1 + 7.42iT - 79T^{2} \) |
| 83 | \( 1 + 9.31T + 83T^{2} \) |
| 89 | \( 1 - 0.418iT - 89T^{2} \) |
| 97 | \( 1 - 2.43iT - 97T^{2} \) |
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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
−9.634293098771160563379977626850, −8.645032118896556432811057142400, −8.306899388302958389205787877787, −6.99531857147751877610840139931, −5.89473151011349848740658270996, −4.78139097802342127876190024829, −3.73662778041994071822886444527, −3.08109821854097978778919265357, −1.65865002449760006242014662737, −1.12701429751422840745182061166,
1.22210892297016557366668286222, 3.66011707223585551842612929798, 4.41567397460630401639908946543, 5.20908262729124926643187672920, 5.96176395478311910955976560782, 6.70659454299442195682026103354, 7.81008864648184783865806804150, 8.455131552215748051769057576619, 8.929168323656434275675118625285, 9.693016923084191302546133306929