L(s) = 1 | + i·2-s − 4-s + (−0.368 − 0.448i)5-s − i·8-s + (0.980 − 0.195i)9-s + (0.448 − 0.368i)10-s + (0.216 − 0.324i)13-s + 16-s + (0.785 + 0.785i)17-s + (0.195 + 0.980i)18-s + (0.368 + 0.448i)20-s + (0.129 − 0.650i)25-s + (0.324 + 0.216i)26-s + (1.38 + 0.275i)29-s + i·32-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s + (−0.368 − 0.448i)5-s − i·8-s + (0.980 − 0.195i)9-s + (0.448 − 0.368i)10-s + (0.216 − 0.324i)13-s + 16-s + (0.785 + 0.785i)17-s + (0.195 + 0.980i)18-s + (0.368 + 0.448i)20-s + (0.129 − 0.650i)25-s + (0.324 + 0.216i)26-s + (1.38 + 0.275i)29-s + i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9535850294\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9535850294\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 257 | \( 1 + T \) |
good | 3 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 5 | \( 1 + (0.368 + 0.448i)T + (-0.195 + 0.980i)T^{2} \) |
| 7 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.216 + 0.324i)T + (-0.382 - 0.923i)T^{2} \) |
| 17 | \( 1 + (-0.785 - 0.785i)T + iT^{2} \) |
| 19 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-1.38 - 0.275i)T + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 37 | \( 1 + (0.512 + 0.273i)T + (0.555 + 0.831i)T^{2} \) |
| 41 | \( 1 + (-0.448 - 1.47i)T + (-0.831 + 0.555i)T^{2} \) |
| 43 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 47 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 53 | \( 1 + (1.47 + 1.21i)T + (0.195 + 0.980i)T^{2} \) |
| 59 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 61 | \( 1 + (-0.425 + 0.636i)T + (-0.382 - 0.923i)T^{2} \) |
| 67 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + (-0.831 + 0.555i)T^{2} \) |
| 73 | \( 1 + (-0.750 + 1.81i)T + (-0.707 - 0.707i)T^{2} \) |
| 79 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 83 | \( 1 + (0.980 + 0.195i)T^{2} \) |
| 89 | \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 97 | \( 1 + (0.151 - 0.124i)T + (0.195 - 0.980i)T^{2} \) |
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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
−10.00467804558299216548528505483, −9.361098434411238862965525944630, −8.203001067291492975309955301124, −7.983296733752818377761147704685, −6.81477499833755512908739883613, −6.17688113524168087259934647558, −5.03094488666880271425307474244, −4.33279426890560608646962687552, −3.35232173063528365131372662894, −1.21394696471838532946309467986,
1.33718969264319444772434139319, 2.68462711100802008411557846431, 3.66412918075816231827281868818, 4.53785579019211182800464160388, 5.46590857692400235798532969497, 6.81528609642138436472278272791, 7.60081222632316254781034802296, 8.516772840452491254031531162415, 9.481581977842810769409623177836, 10.11167941637607208348620059926