L(s) = 1 | − 5-s − 3.12·7-s − 2·11-s − 3.12·13-s − 7.12·17-s − 3.12·19-s − 3.12·23-s + 25-s − 8.24·29-s + 1.12·31-s + 3.12·35-s + 3.12·37-s + 2·41-s + 10.2·43-s + 4.87·47-s + 2.75·49-s − 10·53-s + 2·55-s − 6·59-s − 2·61-s + 3.12·65-s + 10.2·67-s + 8·71-s + 12.2·73-s + 6.24·77-s − 13.1·79-s + 4·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.18·7-s − 0.603·11-s − 0.866·13-s − 1.72·17-s − 0.716·19-s − 0.651·23-s + 0.200·25-s − 1.53·29-s + 0.201·31-s + 0.527·35-s + 0.513·37-s + 0.312·41-s + 1.56·43-s + 0.711·47-s + 0.393·49-s − 1.37·53-s + 0.269·55-s − 0.781·59-s − 0.256·61-s + 0.387·65-s + 1.25·67-s + 0.949·71-s + 1.43·73-s + 0.711·77-s − 1.47·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4554247034\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4554247034\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 1.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.87T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.950757308291772612229333298922, −7.47681181158206434222092298843, −6.61166445034130537682425762194, −6.17421309654511024132675579586, −5.20219443332541596893195210429, −4.34965532793602774129409388465, −3.75588139559950314564638395309, −2.69610938827072509022869345503, −2.13262565291770791757063506624, −0.32930711715944709762783915061,
0.32930711715944709762783915061, 2.13262565291770791757063506624, 2.69610938827072509022869345503, 3.75588139559950314564638395309, 4.34965532793602774129409388465, 5.20219443332541596893195210429, 6.17421309654511024132675579586, 6.61166445034130537682425762194, 7.47681181158206434222092298843, 7.950757308291772612229333298922