L(s) = 1 | + (−0.236 − 0.236i)5-s + 3.27·7-s + (−2.58 + 2.58i)11-s + (1.70 + 1.70i)13-s + 7.05i·17-s + (−3.04 + 3.04i)19-s − 1.47i·23-s − 4.88i·25-s + (2.98 − 2.98i)29-s + 8.02i·31-s + (−0.774 − 0.774i)35-s + (7.93 − 7.93i)37-s + 2.22·41-s + (4.61 + 4.61i)43-s − 7.13·47-s + ⋯ |
L(s) = 1 | + (−0.105 − 0.105i)5-s + 1.23·7-s + (−0.778 + 0.778i)11-s + (0.473 + 0.473i)13-s + 1.71i·17-s + (−0.697 + 0.697i)19-s − 0.307i·23-s − 0.977i·25-s + (0.554 − 0.554i)29-s + 1.44i·31-s + (−0.130 − 0.130i)35-s + (1.30 − 1.30i)37-s + 0.346·41-s + (0.703 + 0.703i)43-s − 1.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.665707988\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.665707988\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.236 + 0.236i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.27T + 7T^{2} \) |
| 11 | \( 1 + (2.58 - 2.58i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.70 - 1.70i)T + 13iT^{2} \) |
| 17 | \( 1 - 7.05iT - 17T^{2} \) |
| 19 | \( 1 + (3.04 - 3.04i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.47iT - 23T^{2} \) |
| 29 | \( 1 + (-2.98 + 2.98i)T - 29iT^{2} \) |
| 31 | \( 1 - 8.02iT - 31T^{2} \) |
| 37 | \( 1 + (-7.93 + 7.93i)T - 37iT^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 + (-4.61 - 4.61i)T + 43iT^{2} \) |
| 47 | \( 1 + 7.13T + 47T^{2} \) |
| 53 | \( 1 + (-5.81 - 5.81i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.46 - 7.46i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.04 - 4.04i)T + 61iT^{2} \) |
| 67 | \( 1 + (-2.90 + 2.90i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.02iT - 71T^{2} \) |
| 73 | \( 1 - 4.08iT - 73T^{2} \) |
| 79 | \( 1 - 5.36iT - 79T^{2} \) |
| 83 | \( 1 + (-3.93 - 3.93i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.35T + 89T^{2} \) |
| 97 | \( 1 - 9.97T + 97T^{2} \) |
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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
−10.16198937674641588591906752954, −8.901375762433178926615093331818, −8.188631888002799843272914892301, −7.73828046739889511658393561061, −6.51523865837199729691295083576, −5.70957278179020946069645729476, −4.55431601514260602600394910297, −4.09316790485268427136031382638, −2.42254978299525239979264893435, −1.47017056441177364479138311661,
0.77940628229878217073227168406, 2.33737002683262009362555469093, 3.33086232544891172105020447314, 4.69453492993779184411972517905, 5.23395279599420287314485425867, 6.26539537620531521273282009523, 7.41857153673886707608897984854, 7.985149675689448183028528068318, 8.745594716995597147777986793220, 9.634663869163551988811184340596