L(s) = 1 | + (−1.40 + 0.118i)2-s + (1.97 − 0.335i)4-s + 3.46i·5-s + 7-s + (−2.73 + 0.707i)8-s + (−0.411 − 4.88i)10-s − 3.31i·11-s − 3.10i·13-s + (−1.40 + 0.118i)14-s + (3.77 − 1.32i)16-s + 1.40·17-s − 4.80i·19-s + (1.16 + 6.82i)20-s + (0.394 + 4.67i)22-s − 8.79·23-s + ⋯ |
L(s) = 1 | + (−0.996 + 0.0841i)2-s + (0.985 − 0.167i)4-s + 1.54i·5-s + 0.377·7-s + (−0.968 + 0.249i)8-s + (−0.130 − 1.54i)10-s − 1.00i·11-s − 0.862i·13-s + (−0.376 + 0.0317i)14-s + (0.943 − 0.330i)16-s + 0.340·17-s − 1.10i·19-s + (0.259 + 1.52i)20-s + (0.0841 + 0.997i)22-s − 1.83·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6660891606\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6660891606\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.118i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 3.31iT - 11T^{2} \) |
| 13 | \( 1 + 3.10iT - 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 + 4.80iT - 19T^{2} \) |
| 23 | \( 1 + 8.79T + 23T^{2} \) |
| 29 | \( 1 + 9.87iT - 29T^{2} \) |
| 31 | \( 1 + 7.83T + 31T^{2} \) |
| 37 | \( 1 + 5.42iT - 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 7.03iT - 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6.51iT - 53T^{2} \) |
| 59 | \( 1 + 3.89iT - 59T^{2} \) |
| 61 | \( 1 - 9.42iT - 61T^{2} \) |
| 67 | \( 1 - 0.909iT - 67T^{2} \) |
| 71 | \( 1 - 6.42T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 + 0.370iT - 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−9.409737294485442637552236172492, −8.364709346596747027171640183063, −7.71301584382030285228323784859, −7.13891510500186505496268987597, −6.05431443419561598601421626332, −5.73386407818846581483823609511, −3.88622261305751196215819124216, −2.91639927383210131644996732128, −2.15206964189892734735405585052, −0.35971477758597765316399104843,
1.41379310254297428079622939298, 1.92522915630097362714938372081, 3.69210532493531815849670572869, 4.64997617561910884144595544115, 5.55516586117261252156861213532, 6.52720684748974526556180881799, 7.61458574696717445647773101041, 8.113039080631577431200356652079, 8.891000091110648568699728813198, 9.571119231474322397831216473097