| L(s) = 1 | − i·2-s − 4-s − i·5-s − 4.44·7-s + i·8-s − 10-s + 2.44·11-s + 0.440i·13-s + 4.44i·14-s + 16-s + 4.83i·17-s − 0.440i·19-s + i·20-s − 2.44i·22-s + 2.83i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447i·5-s − 1.67·7-s + 0.353i·8-s − 0.316·10-s + 0.735·11-s + 0.122i·13-s + 1.18i·14-s + 0.250·16-s + 1.17i·17-s − 0.101i·19-s + 0.223i·20-s − 0.520i·22-s + 0.591i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0326 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0326 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.193241748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.193241748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (0.198 + 6.07i)T \) |
| good | 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 - 0.440iT - 13T^{2} \) |
| 17 | \( 1 - 4.83iT - 17T^{2} \) |
| 19 | \( 1 + 0.440iT - 19T^{2} \) |
| 23 | \( 1 - 2.83iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 5.27iT - 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 0.396iT - 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 + 2.39iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 9.67T + 71T^{2} \) |
| 73 | \( 1 - 5.71T + 73T^{2} \) |
| 79 | \( 1 - 13.2iT - 79T^{2} \) |
| 83 | \( 1 - 7.71T + 83T^{2} \) |
| 89 | \( 1 + 15.3iT - 89T^{2} \) |
| 97 | \( 1 + 2.88iT - 97T^{2} \) |
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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
−8.797005116680725972819551458776, −7.77729842621794679443693799801, −6.87133353601978709938833528511, −6.09829433452542619384380031505, −5.50953314357145202158056834662, −4.15653786900666020914678084547, −3.79564800919907901034906476791, −2.85834565546965752520392073896, −1.77605053565689213495043046846, −0.54357798527110780444767136925,
0.76510851797553392525648481286, 2.57015074008354329430528773611, 3.32726519410233554423272152097, 4.11722507872913479241489191305, 5.14519216174186025164541392810, 6.08897962645012952337648654632, 6.59453053249528340199618894445, 7.09572592324136605552705436621, 7.901930343678904531479005146944, 9.085203680721936783488252622501
