L(s) = 1 | + 2-s + (−0.571 + 1.63i)3-s + 4-s + (2.12 + 0.689i)5-s + (−0.571 + 1.63i)6-s + 1.28·7-s + 8-s + (−2.34 − 1.86i)9-s + (2.12 + 0.689i)10-s + 4.52·11-s + (−0.571 + 1.63i)12-s − 3.04i·13-s + 1.28·14-s + (−2.34 + 3.08i)15-s + 16-s + 5.73i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.330 + 0.943i)3-s + 0.5·4-s + (0.951 + 0.308i)5-s + (−0.233 + 0.667i)6-s + 0.487·7-s + 0.353·8-s + (−0.781 − 0.623i)9-s + (0.672 + 0.217i)10-s + 1.36·11-s + (−0.165 + 0.471i)12-s − 0.843i·13-s + 0.344·14-s + (−0.605 + 0.796i)15-s + 0.250·16-s + 1.39i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.31168 + 1.18672i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.31168 + 1.18672i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.571 - 1.63i)T \) |
| 5 | \( 1 + (-2.12 - 0.689i)T \) |
| 23 | \( 1 + (4.79 - 0.132i)T \) |
good | 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 + 3.04iT - 13T^{2} \) |
| 17 | \( 1 - 5.73iT - 17T^{2} \) |
| 19 | \( 1 + 3.52iT - 19T^{2} \) |
| 29 | \( 1 + 3.03iT - 29T^{2} \) |
| 31 | \( 1 + 7.30T + 31T^{2} \) |
| 37 | \( 1 + 8.69T + 37T^{2} \) |
| 41 | \( 1 - 12.7iT - 41T^{2} \) |
| 43 | \( 1 - 8.85T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 - 4.54iT - 53T^{2} \) |
| 59 | \( 1 + 12.7iT - 59T^{2} \) |
| 61 | \( 1 + 3.04iT - 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 - 5.30iT - 71T^{2} \) |
| 73 | \( 1 + 9.43iT - 73T^{2} \) |
| 79 | \( 1 + 16.0iT - 79T^{2} \) |
| 83 | \( 1 - 8.09iT - 83T^{2} \) |
| 89 | \( 1 - 6.63T + 89T^{2} \) |
| 97 | \( 1 - 0.0268T + 97T^{2} \) |
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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
−10.73921207634754716012377217982, −9.867591436799213856511834308703, −9.116572796680071092246475925600, −8.051314867227601250653515044633, −6.58150743009319674012490028449, −6.01945560734223926298874175457, −5.16367165771919826056475404622, −4.15102847919531432924546726787, −3.23706631634625917575326113308, −1.74369059001863493916388962545,
1.42869697951285531602527069739, 2.21934794140734710437089252415, 3.86684720339589132465050804680, 5.10926282835041791436677307548, 5.80700347515188181751778775832, 6.72661622986385665703493984962, 7.33135669828972981989563109144, 8.656156295340321082978173368609, 9.367110816555509287227061537896, 10.55952503044415476759112742067