L(s) = 1 | − 2.66·2-s − 3-s + 5.12·4-s + 5-s + 2.66·6-s − 3.27·7-s − 8.35·8-s + 9-s − 2.66·10-s − 0.898·11-s − 5.12·12-s + 13-s + 8.75·14-s − 15-s + 12.0·16-s − 0.0313·17-s − 2.66·18-s − 1.43·19-s + 5.12·20-s + 3.27·21-s + 2.39·22-s + 0.246·23-s + 8.35·24-s + 25-s − 2.66·26-s − 27-s − 16.8·28-s + ⋯ |
L(s) = 1 | − 1.88·2-s − 0.577·3-s + 2.56·4-s + 0.447·5-s + 1.08·6-s − 1.23·7-s − 2.95·8-s + 0.333·9-s − 0.844·10-s − 0.270·11-s − 1.48·12-s + 0.277·13-s + 2.33·14-s − 0.258·15-s + 3.01·16-s − 0.00759·17-s − 0.629·18-s − 0.330·19-s + 1.14·20-s + 0.715·21-s + 0.511·22-s + 0.0514·23-s + 1.70·24-s + 0.200·25-s − 0.523·26-s − 0.192·27-s − 3.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 7 | \( 1 + 3.27T + 7T^{2} \) |
| 11 | \( 1 + 0.898T + 11T^{2} \) |
| 17 | \( 1 + 0.0313T + 17T^{2} \) |
| 19 | \( 1 + 1.43T + 19T^{2} \) |
| 23 | \( 1 - 0.246T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 6.72T + 41T^{2} \) |
| 43 | \( 1 + 6.43T + 43T^{2} \) |
| 47 | \( 1 + 0.852T + 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 - 7.71T + 59T^{2} \) |
| 61 | \( 1 - 9.88T + 61T^{2} \) |
| 67 | \( 1 - 5.92T + 67T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 6.01T + 79T^{2} \) |
| 83 | \( 1 - 5.52T + 83T^{2} \) |
| 89 | \( 1 + 0.0871T + 89T^{2} \) |
| 97 | \( 1 - 1.31T + 97T^{2} \) |
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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
−7.81984554398316060812710595343, −6.90454804840297777471464338423, −6.66573026313489214652292813355, −5.96491248316030181641451294101, −5.19581222728931718608693718989, −3.74302759075060282165888085738, −2.84141107362620074597326697247, −2.01639507414378792467138288507, −0.953486337971263440291608297175, 0,
0.953486337971263440291608297175, 2.01639507414378792467138288507, 2.84141107362620074597326697247, 3.74302759075060282165888085738, 5.19581222728931718608693718989, 5.96491248316030181641451294101, 6.66573026313489214652292813355, 6.90454804840297777471464338423, 7.81984554398316060812710595343