L(s) = 1 | − 3-s + 5-s + 4.76·7-s + 9-s − 5.60·11-s + 3.60·13-s − 15-s − 4.12·17-s − 2.64·19-s − 4.76·21-s + 23-s + 25-s − 27-s + 3.87·29-s − 9.79·31-s + 5.60·33-s + 4.76·35-s − 6.76·37-s − 3.60·39-s − 11.3·41-s + 2.96·43-s + 45-s − 1.35·47-s + 15.7·49-s + 4.12·51-s − 3.09·53-s − 5.60·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.80·7-s + 0.333·9-s − 1.69·11-s + 1.00·13-s − 0.258·15-s − 1.00·17-s − 0.605·19-s − 1.03·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.719·29-s − 1.75·31-s + 0.976·33-s + 0.805·35-s − 1.11·37-s − 0.578·39-s − 1.77·41-s + 0.452·43-s + 0.149·45-s − 0.198·47-s + 2.24·49-s + 0.577·51-s − 0.425·53-s − 0.756·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 - 4.76T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 9.79T + 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 + 3.09T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + 6.82T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 8.31T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−7.74112822221903717081373525517, −7.21142313550157766888958657005, −6.19747148837246034280457361880, −5.55699781568931905628552777582, −4.87837771860576898463518947924, −4.48435171572504983737314192155, −3.21810720804624491943046183880, −2.04233665602397301838800066253, −1.53949375176441335385279691145, 0,
1.53949375176441335385279691145, 2.04233665602397301838800066253, 3.21810720804624491943046183880, 4.48435171572504983737314192155, 4.87837771860576898463518947924, 5.55699781568931905628552777582, 6.19747148837246034280457361880, 7.21142313550157766888958657005, 7.74112822221903717081373525517