| L(s) = 1 | + 3i·3-s + 8i·5-s + 12·7-s − 9·9-s + 12i·11-s + 20i·13-s − 24·15-s + 62·17-s + 108i·19-s + 36i·21-s − 72·23-s + 61·25-s − 27i·27-s − 128i·29-s − 204·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.715i·5-s + 0.647·7-s − 0.333·9-s + 0.328i·11-s + 0.426i·13-s − 0.413·15-s + 0.884·17-s + 1.30i·19-s + 0.374i·21-s − 0.652·23-s + 0.487·25-s − 0.192i·27-s − 0.819i·29-s − 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.603145310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.603145310\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| good | 5 | \( 1 - 8iT - 125T^{2} \) |
| 7 | \( 1 - 12T + 343T^{2} \) |
| 11 | \( 1 - 12iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 62T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 72T + 1.21e4T^{2} \) |
| 29 | \( 1 + 128iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 204T + 2.97e4T^{2} \) |
| 37 | \( 1 - 228iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 22T + 6.89e4T^{2} \) |
| 43 | \( 1 - 204iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 600T + 1.03e5T^{2} \) |
| 53 | \( 1 + 256iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 828iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 84iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 348iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 456T + 3.57e5T^{2} \) |
| 73 | \( 1 - 822T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.35e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 108iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 938T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.27e3T + 9.12e5T^{2} \) |
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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
−11.25320203167392078872742371362, −10.22665350304568739322843965755, −9.723481108191528504881255951692, −8.404868040358122708562215616763, −7.63449617440326613744674874774, −6.45355276658753861853258837915, −5.40228291509979591401147309975, −4.26240616441031676735322052055, −3.19128049737484532564874046418, −1.71040998256064115654498497226,
0.54872275098143309599137984574, 1.78702475744060968774524541995, 3.31499842296347331767532362132, 4.83957795552222977033539825418, 5.58354550709204744473542521018, 6.87394291216297848290382686869, 7.86072404831049335806811520909, 8.597579411705327839531431692221, 9.489656622657277754922864419143, 10.78179207922824643398379039771
