L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−1.25 − 1.84i)5-s + (3.10 + 3.10i)7-s + (0.707 + 0.707i)8-s + (2.19 + 0.416i)10-s + 3.82·11-s + (−0.0891 − 0.0891i)13-s − 4.39·14-s − 1.00·16-s + (1.83 + 1.83i)17-s + (−2.70 + 3.41i)19-s + (−1.84 + 1.25i)20-s + (−2.70 + 2.70i)22-s + (−2.58 + 2.58i)23-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.562 − 0.826i)5-s + (1.17 + 1.17i)7-s + (0.250 + 0.250i)8-s + (0.694 + 0.131i)10-s + 1.15·11-s + (−0.0247 − 0.0247i)13-s − 1.17·14-s − 0.250·16-s + (0.443 + 0.443i)17-s + (−0.620 + 0.784i)19-s + (−0.413 + 0.281i)20-s + (−0.576 + 0.576i)22-s + (−0.539 + 0.539i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.353343147\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.353343147\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (1.25 + 1.84i)T \) |
| 19 | \( 1 + (2.70 - 3.41i)T \) |
good | 7 | \( 1 + (-3.10 - 3.10i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 + (0.0891 + 0.0891i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.83 - 1.83i)T + 17iT^{2} \) |
| 23 | \( 1 + (2.58 - 2.58i)T - 23iT^{2} \) |
| 29 | \( 1 + 3.60T + 29T^{2} \) |
| 31 | \( 1 + 3.60iT - 31T^{2} \) |
| 37 | \( 1 + (-7.07 + 7.07i)T - 37iT^{2} \) |
| 41 | \( 1 - 11.2iT - 41T^{2} \) |
| 43 | \( 1 + (-7.93 + 7.93i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.463 - 0.463i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.21 - 3.21i)T + 53iT^{2} \) |
| 59 | \( 1 + 9.40T + 59T^{2} \) |
| 61 | \( 1 - 8.21T + 61T^{2} \) |
| 67 | \( 1 + (8.78 - 8.78i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.66iT - 71T^{2} \) |
| 73 | \( 1 + (3.64 - 3.64i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.82T + 79T^{2} \) |
| 83 | \( 1 + (-0.347 + 0.347i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + (6.88 - 6.88i)T - 97iT^{2} \) |
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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
−9.100883769941452001470102067375, −8.763524864924275264780318295438, −7.934355803028394811448818457251, −7.49435813977884403050361491426, −5.99821587828937969740004792248, −5.70914562442925123793327582195, −4.57764354536461672281012182701, −3.88149941322457891021153596818, −2.10838536433786170191173082839, −1.21416654337134159742497677905,
0.71333945446280298905810409105, 1.90960684210304095978044664807, 3.15943115784256906026885532842, 4.13183217790453095190723572272, 4.60979625249339810184249064835, 6.20253768937063035986851610104, 7.10099137369005077577957724861, 7.56541045549384415348147988584, 8.352937384275427377271383673788, 9.182163418288491467560271143594