| L(s) = 1 | − 2-s + 4-s − 0.815·5-s − 4.13·7-s − 8-s + 0.815·10-s − 5.38·11-s + 5.53·13-s + 4.13·14-s + 16-s + 1.97·17-s − 7.77·19-s − 0.815·20-s + 5.38·22-s + 4.49·23-s − 4.33·25-s − 5.53·26-s − 4.13·28-s + 4.21·29-s − 6.64·31-s − 32-s − 1.97·34-s + 3.36·35-s − 1.33·37-s + 7.77·38-s + 0.815·40-s − 4.44·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.364·5-s − 1.56·7-s − 0.353·8-s + 0.257·10-s − 1.62·11-s + 1.53·13-s + 1.10·14-s + 0.250·16-s + 0.479·17-s − 1.78·19-s − 0.182·20-s + 1.14·22-s + 0.937·23-s − 0.866·25-s − 1.08·26-s − 0.780·28-s + 0.782·29-s − 1.19·31-s − 0.176·32-s − 0.339·34-s + 0.569·35-s − 0.219·37-s + 1.26·38-s + 0.128·40-s − 0.693·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3417761840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3417761840\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 0.815T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 5.38T + 11T^{2} \) |
| 13 | \( 1 - 5.53T + 13T^{2} \) |
| 17 | \( 1 - 1.97T + 17T^{2} \) |
| 19 | \( 1 + 7.77T + 19T^{2} \) |
| 23 | \( 1 - 4.49T + 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 + 6.64T + 31T^{2} \) |
| 37 | \( 1 + 1.33T + 37T^{2} \) |
| 41 | \( 1 + 4.44T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 3.59T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 - 1.83T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 - 1.93T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 8.98T + 97T^{2} \) |
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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
−8.041742596485428838519103259299, −7.09179303321055641689889189470, −6.57209325719866224103821065154, −5.94081098350008170591134031558, −5.23117105726302877868295200574, −4.04838399636665285237282271606, −3.33917049048588975495663498583, −2.75879870986026898976914354250, −1.68082733440976496654047991185, −0.30979173155962321095448441725,
0.30979173155962321095448441725, 1.68082733440976496654047991185, 2.75879870986026898976914354250, 3.33917049048588975495663498583, 4.04838399636665285237282271606, 5.23117105726302877868295200574, 5.94081098350008170591134031558, 6.57209325719866224103821065154, 7.09179303321055641689889189470, 8.041742596485428838519103259299
