L(s) = 1 | − 2-s + 4-s + 5-s − 4.59·7-s − 8-s − 10-s − 5.13·11-s − 1.22·13-s + 4.59·14-s + 16-s + 4.68·17-s − 4.59·19-s + 20-s + 5.13·22-s + 23-s + 25-s + 1.22·26-s − 4.59·28-s − 3.37·29-s − 0.777·31-s − 32-s − 4.68·34-s − 4.59·35-s + 5.81·37-s + 4.59·38-s − 40-s + 8.50·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 1.73·7-s − 0.353·8-s − 0.316·10-s − 1.54·11-s − 0.338·13-s + 1.22·14-s + 0.250·16-s + 1.13·17-s − 1.05·19-s + 0.223·20-s + 1.09·22-s + 0.208·23-s + 0.200·25-s + 0.239·26-s − 0.868·28-s − 0.626·29-s − 0.139·31-s − 0.176·32-s − 0.803·34-s − 0.777·35-s + 0.956·37-s + 0.745·38-s − 0.158·40-s + 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7561824048\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7561824048\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 4.59T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 + 4.59T + 19T^{2} \) |
| 29 | \( 1 + 3.37T + 29T^{2} \) |
| 31 | \( 1 + 0.777T + 31T^{2} \) |
| 37 | \( 1 - 5.81T + 37T^{2} \) |
| 41 | \( 1 - 8.50T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 6.44T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 9.37T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 15.6T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 + 4.44T + 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 + 8.93T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−9.417246562727572730971649577342, −8.375757539500270428583267452795, −7.56456030324053744079270194696, −6.90897266533509030199053305931, −5.92895976119256469435791416814, −5.52327010564622602133984201579, −4.05464090548458572120320279292, −2.90532806023335387889832660096, −2.38399835308151187172869194962, −0.60212299356919442244351812269,
0.60212299356919442244351812269, 2.38399835308151187172869194962, 2.90532806023335387889832660096, 4.05464090548458572120320279292, 5.52327010564622602133984201579, 5.92895976119256469435791416814, 6.90897266533509030199053305931, 7.56456030324053744079270194696, 8.375757539500270428583267452795, 9.417246562727572730971649577342