| L(s) = 1 | + 3.21·3-s + 1.89·5-s + 7-s + 7.34·9-s + 11-s − 13-s + 6.11·15-s + 4.99·17-s + 5.59·19-s + 3.21·21-s − 0.631·23-s − 1.39·25-s + 13.9·27-s − 0.143·29-s − 4.47·31-s + 3.21·33-s + 1.89·35-s − 1.35·37-s − 3.21·39-s + 0.221·41-s − 7.36·43-s + 13.9·45-s + 0.631·47-s + 49-s + 16.0·51-s − 3.13·53-s + 1.89·55-s + ⋯ |
| L(s) = 1 | + 1.85·3-s + 0.849·5-s + 0.377·7-s + 2.44·9-s + 0.301·11-s − 0.277·13-s + 1.57·15-s + 1.21·17-s + 1.28·19-s + 0.701·21-s − 0.131·23-s − 0.278·25-s + 2.69·27-s − 0.0266·29-s − 0.804·31-s + 0.559·33-s + 0.321·35-s − 0.222·37-s − 0.515·39-s + 0.0345·41-s − 1.12·43-s + 2.08·45-s + 0.0921·47-s + 0.142·49-s + 2.25·51-s − 0.431·53-s + 0.256·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.036947294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.036947294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.21T + 3T^{2} \) |
| 5 | \( 1 - 1.89T + 5T^{2} \) |
| 17 | \( 1 - 4.99T + 17T^{2} \) |
| 19 | \( 1 - 5.59T + 19T^{2} \) |
| 23 | \( 1 + 0.631T + 23T^{2} \) |
| 29 | \( 1 + 0.143T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 1.35T + 37T^{2} \) |
| 41 | \( 1 - 0.221T + 41T^{2} \) |
| 43 | \( 1 + 7.36T + 43T^{2} \) |
| 47 | \( 1 - 0.631T + 47T^{2} \) |
| 53 | \( 1 + 3.13T + 53T^{2} \) |
| 59 | \( 1 - 0.632T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 - 3.15T + 67T^{2} \) |
| 71 | \( 1 + 8.21T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 - 8.15T + 79T^{2} \) |
| 83 | \( 1 + 0.585T + 83T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83913907371694310906578258310, −7.42670510785394580859193032197, −6.67887822338776724737634608669, −5.64273585754100694248305830266, −5.02755463179069511132756230169, −4.00925200528312696709780299109, −3.36759895144617195482822200008, −2.71460245586072829520150742639, −1.83077405882630584711332064474, −1.28533769802837807158531066390,
1.28533769802837807158531066390, 1.83077405882630584711332064474, 2.71460245586072829520150742639, 3.36759895144617195482822200008, 4.00925200528312696709780299109, 5.02755463179069511132756230169, 5.64273585754100694248305830266, 6.67887822338776724737634608669, 7.42670510785394580859193032197, 7.83913907371694310906578258310
