| L(s) = 1 | − 0.184i·2-s − i·3-s + 1.96·4-s + (1.94 − 1.10i)5-s − 0.184·6-s − 1.90i·7-s − 0.733i·8-s − 9-s + (−0.204 − 0.359i)10-s − 3.94·11-s − 1.96i·12-s + 4.53i·13-s − 0.353·14-s + (−1.10 − 1.94i)15-s + 3.79·16-s + 4.68i·17-s + ⋯ |
| L(s) = 1 | − 0.130i·2-s − 0.577i·3-s + 0.982·4-s + (0.868 − 0.495i)5-s − 0.0754·6-s − 0.721i·7-s − 0.259i·8-s − 0.333·9-s + (−0.0648 − 0.113i)10-s − 1.18·11-s − 0.567i·12-s + 1.25i·13-s − 0.0943·14-s + (−0.286 − 0.501i)15-s + 0.949·16-s + 1.13i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45152 - 0.842833i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45152 - 0.842833i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1.94 + 1.10i)T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.184iT - 2T^{2} \) |
| 7 | \( 1 + 1.90iT - 7T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 - 4.53iT - 13T^{2} \) |
| 17 | \( 1 - 4.68iT - 17T^{2} \) |
| 23 | \( 1 + 6.18iT - 23T^{2} \) |
| 29 | \( 1 + 5.98T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 - 8.07iT - 37T^{2} \) |
| 41 | \( 1 - 0.0567T + 41T^{2} \) |
| 43 | \( 1 - 0.822iT - 43T^{2} \) |
| 47 | \( 1 - 7.50iT - 47T^{2} \) |
| 53 | \( 1 - 1.28iT - 53T^{2} \) |
| 59 | \( 1 - 8.28T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 10.0iT - 73T^{2} \) |
| 79 | \( 1 + 4.12T + 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + 7.17T + 89T^{2} \) |
| 97 | \( 1 + 2.70iT - 97T^{2} \) |
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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
−11.71972115655645116481487283752, −10.63843148565632705477337699483, −10.09815855152892004508403563156, −8.691703426777020269275340651572, −7.71013611101382487274142424411, −6.64295529191528763639294665415, −5.95481395097019065238080075985, −4.47319398171964136917146872382, −2.63305018453531703324147097809, −1.52738499713868905304444150540,
2.34919869762140200109016484605, 3.12768567073207124784425312770, 5.42858186997531344168050673643, 5.65743580084348665626072517239, 7.09965198256650001444760400794, 8.026002219533849611747956580322, 9.360674052159638163056487176014, 10.23599351478773688722605842218, 10.88632589760258277856595146558, 11.79572798982419347473985115491
