L(s) = 1 | − 4.27·5-s + 4.77i·7-s − 3·9-s + 2.15i·11-s + 0.274·17-s − 4.35i·19-s − 8.71i·23-s + 13.2·25-s − 20.4i·35-s + 7.40i·43-s + 12.8·45-s − 9.07i·47-s − 15.8·49-s − 9.19i·55-s + 3.72·61-s + ⋯ |
L(s) = 1 | − 1.91·5-s + 1.80i·7-s − 9-s + 0.648i·11-s + 0.0666·17-s − 0.999i·19-s − 1.81i·23-s + 2.65·25-s − 3.45i·35-s + 1.12i·43-s + 1.91·45-s − 1.32i·47-s − 2.26·49-s − 1.23i·55-s + 0.476·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3151947294\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3151947294\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 4.27T + 5T^{2} \) |
| 7 | \( 1 - 4.77iT - 7T^{2} \) |
| 11 | \( 1 - 2.15iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 0.274T + 17T^{2} \) |
| 23 | \( 1 + 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 7.40iT - 43T^{2} \) |
| 47 | \( 1 + 9.07iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 3.72T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.160486899394893611915655681478, −8.599305754364492006218002211911, −8.153796937980897713385019665524, −7.11976562377767559781329086965, −6.22874240249406335119383773593, −5.11068542833249976671561847951, −4.43211591071525013938648238598, −3.14026589385116810095749408704, −2.46721256464687153039851315276, −0.16788113180149400012379331689,
1.01095105835856636341406806106, 3.31322421222792031243241309871, 3.66776447146328898094091819872, 4.52393997958540853495981910957, 5.71527688299552490967522665542, 6.93169628705816716995807753944, 7.64781684464786081910432226637, 8.021907053161617209527506853681, 8.916929548263368977706596870749, 10.13032840776092542420332590181