| L(s) = 1 | + 0.414·2-s − 1.82·4-s + 5-s − 3.82·7-s − 1.58·8-s + 0.414·10-s + 1.82·11-s − 13-s − 1.58·14-s + 3·16-s + 6.65·17-s − 6.82·19-s − 1.82·20-s + 0.757·22-s − 5.65·23-s + 25-s − 0.414·26-s + 7·28-s + 5.65·29-s + 6·31-s + 4.41·32-s + 2.75·34-s − 3.82·35-s + 6·37-s − 2.82·38-s − 1.58·40-s − 5.82·41-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s + 0.447·5-s − 1.44·7-s − 0.560·8-s + 0.130·10-s + 0.551·11-s − 0.277·13-s − 0.423·14-s + 0.750·16-s + 1.61·17-s − 1.56·19-s − 0.408·20-s + 0.161·22-s − 1.17·23-s + 0.200·25-s − 0.0812·26-s + 1.32·28-s + 1.05·29-s + 1.07·31-s + 0.780·32-s + 0.472·34-s − 0.647·35-s + 0.986·37-s − 0.458·38-s − 0.250·40-s − 0.910·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.297260608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.297260608\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 5.82T + 41T^{2} \) |
| 43 | \( 1 + 3.65T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 5.48T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 9T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−9.408837326530847529328397975738, −8.632481283272894220303342042887, −7.85982138595939678015041299631, −6.53630172867317970577809673734, −6.17866097524127883999957234803, −5.27062369713343111927331056428, −4.19467387188135897283540979590, −3.52360163980507567003309290969, −2.49438362921991853900040886749, −0.74437641420093412151134754458,
0.74437641420093412151134754458, 2.49438362921991853900040886749, 3.52360163980507567003309290969, 4.19467387188135897283540979590, 5.27062369713343111927331056428, 6.17866097524127883999957234803, 6.53630172867317970577809673734, 7.85982138595939678015041299631, 8.632481283272894220303342042887, 9.408837326530847529328397975738
