L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.0951 − 0.0951i)5-s + (−0.0951 + 2.64i)7-s + (−0.707 + 0.707i)8-s + 0.134·10-s + (3.64 − 3.64i)11-s + (−2 − 3i)13-s + (−1.93 + 1.80i)14-s − 1.00·16-s + 5.98·17-s + (4.19 − 4.19i)19-s + (0.0951 + 0.0951i)20-s + 5.15·22-s + 4.69i·23-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.0425 − 0.0425i)5-s + (−0.0359 + 0.999i)7-s + (−0.250 + 0.250i)8-s + 0.0425·10-s + (1.09 − 1.09i)11-s + (−0.554 − 0.832i)13-s + (−0.517 + 0.481i)14-s − 0.250·16-s + 1.45·17-s + (0.961 − 0.961i)19-s + (0.0212 + 0.0212i)20-s + 1.09·22-s + 0.978i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.472106449\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.472106449\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.0951 - 2.64i)T \) |
| 13 | \( 1 + (2 + 3i)T \) |
good | 5 | \( 1 + (-0.0951 + 0.0951i)T - 5iT^{2} \) |
| 11 | \( 1 + (-3.64 + 3.64i)T - 11iT^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 + (-4.19 + 4.19i)T - 19iT^{2} \) |
| 23 | \( 1 - 4.69iT - 23T^{2} \) |
| 29 | \( 1 - 4.59T + 29T^{2} \) |
| 31 | \( 1 + (-0.739 + 0.739i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.83 - 4.83i)T - 37iT^{2} \) |
| 41 | \( 1 + (-3.04 + 3.04i)T - 41iT^{2} \) |
| 43 | \( 1 - 8.78iT - 43T^{2} \) |
| 47 | \( 1 + (-3.28 - 3.28i)T + 47iT^{2} \) |
| 53 | \( 1 + 1.09T + 53T^{2} \) |
| 59 | \( 1 + (-1.30 - 1.30i)T + 59iT^{2} \) |
| 61 | \( 1 - 5.98iT - 61T^{2} \) |
| 67 | \( 1 + (0.0454 + 0.0454i)T + 67iT^{2} \) |
| 71 | \( 1 + (8.69 + 8.69i)T + 71iT^{2} \) |
| 73 | \( 1 + (-4.83 - 4.83i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + (1.28 - 1.28i)T - 83iT^{2} \) |
| 89 | \( 1 + (9.21 + 9.21i)T + 89iT^{2} \) |
| 97 | \( 1 + (-9.97 + 9.97i)T - 97iT^{2} \) |
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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
−9.325095619038582223891087406198, −8.714061439082116513645342485637, −7.82700705903677534767893794289, −7.12392929307141623505976098918, −5.98501779993579133481756268983, −5.60246003206257761707474825550, −4.77293500473058498089864752023, −3.33420847328821095255714831804, −2.98465944112659061804362975487, −1.19225277927141067179385326964,
1.02601203525348156533897041156, 2.07149319723641651976015799522, 3.41764802111673414373567356664, 4.17235366566764480489407449161, 4.84083651636627609653229950696, 5.97752510540121677754604876570, 6.91030684054044631226834937736, 7.39816718205696807440111565188, 8.521371565482420544063801005634, 9.658525908155263882406704570567