L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (2.56 + 2.56i)5-s + (−0.648 + 2.56i)7-s + (0.707 + 0.707i)8-s − 3.62·10-s + (1.64 + 1.64i)11-s + (2 − 3i)13-s + (−1.35 − 2.27i)14-s − 1.00·16-s + 7.15·17-s + (−2.86 − 2.86i)19-s + (2.56 − 2.56i)20-s − 2.33·22-s + 4.45i·23-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (1.14 + 1.14i)5-s + (−0.245 + 0.969i)7-s + (0.250 + 0.250i)8-s − 1.14·10-s + (0.496 + 0.496i)11-s + (0.554 − 0.832i)13-s + (−0.362 − 0.607i)14-s − 0.250·16-s + 1.73·17-s + (−0.656 − 0.656i)19-s + (0.573 − 0.573i)20-s − 0.496·22-s + 0.929i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.773200918\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.773200918\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.648 - 2.56i)T \) |
| 13 | \( 1 + (-2 + 3i)T \) |
good | 5 | \( 1 + (-2.56 - 2.56i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.64 - 1.64i)T + 11iT^{2} \) |
| 17 | \( 1 - 7.15T + 17T^{2} \) |
| 19 | \( 1 + (2.86 + 2.86i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.45iT - 23T^{2} \) |
| 29 | \( 1 - 9.75T + 29T^{2} \) |
| 31 | \( 1 + (-3.91 - 3.91i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.48 - 2.48i)T + 37iT^{2} \) |
| 41 | \( 1 + (7.53 + 7.53i)T + 41iT^{2} \) |
| 43 | \( 1 + 8.62iT - 43T^{2} \) |
| 47 | \( 1 + (-0.703 + 0.703i)T - 47iT^{2} \) |
| 53 | \( 1 + 2.42T + 53T^{2} \) |
| 59 | \( 1 + (10.4 - 10.4i)T - 59iT^{2} \) |
| 61 | \( 1 + 7.15iT - 61T^{2} \) |
| 67 | \( 1 + (4.53 - 4.53i)T - 67iT^{2} \) |
| 71 | \( 1 + (-0.456 + 0.456i)T - 71iT^{2} \) |
| 73 | \( 1 + (-2.48 + 2.48i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + (2.70 + 2.70i)T + 83iT^{2} \) |
| 89 | \( 1 + (4.75 - 4.75i)T - 89iT^{2} \) |
| 97 | \( 1 + (4.49 + 4.49i)T + 97iT^{2} \) |
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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
−9.656979283502137282455226781454, −8.869195636067037287793690411260, −8.072756617404897071442470001537, −7.04229239282473412044834723096, −6.41239525554318920191616441926, −5.77651636891089433752340019786, −5.08416950458202816933338074875, −3.35873264340533555604377052869, −2.59146125104822342816887649837, −1.42080425068235545030891416407,
0.952224231794907351857241316121, 1.52886656984306089881299053865, 2.97991403562017103984853117465, 4.11225569267437810288602162356, 4.83519010924601327283367528608, 6.14687137327945816296143946283, 6.50182154861164117767861415773, 7.980385580733234083739904518631, 8.394125136458710488149199885161, 9.353771171915455461241537224853