| L(s) = 1 | − 3-s + 5·5-s − 7·7-s − 26·9-s − 65·11-s − 13·13-s − 5·15-s − 73·17-s − 142·19-s + 7·21-s − 130·23-s + 25·25-s + 53·27-s − 111·29-s − 256·31-s + 65·33-s − 35·35-s + 266·37-s + 13·39-s − 424·41-s + 534·43-s − 130·45-s + 269·47-s + 49·49-s + 73·51-s + 132·53-s − 325·55-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.447·5-s − 0.377·7-s − 0.962·9-s − 1.78·11-s − 0.277·13-s − 0.0860·15-s − 1.04·17-s − 1.71·19-s + 0.0727·21-s − 1.17·23-s + 1/5·25-s + 0.377·27-s − 0.710·29-s − 1.48·31-s + 0.342·33-s − 0.169·35-s + 1.18·37-s + 0.0533·39-s − 1.61·41-s + 1.89·43-s − 0.430·45-s + 0.834·47-s + 1/7·49-s + 0.200·51-s + 0.342·53-s − 0.796·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2689729650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2689729650\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 11 | \( 1 + 65 T + p^{3} T^{2} \) |
| 13 | \( 1 + p T + p^{3} T^{2} \) |
| 17 | \( 1 + 73 T + p^{3} T^{2} \) |
| 19 | \( 1 + 142 T + p^{3} T^{2} \) |
| 23 | \( 1 + 130 T + p^{3} T^{2} \) |
| 29 | \( 1 + 111 T + p^{3} T^{2} \) |
| 31 | \( 1 + 256 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 + 424 T + p^{3} T^{2} \) |
| 43 | \( 1 - 534 T + p^{3} T^{2} \) |
| 47 | \( 1 - 269 T + p^{3} T^{2} \) |
| 53 | \( 1 - 132 T + p^{3} T^{2} \) |
| 59 | \( 1 + 224 T + p^{3} T^{2} \) |
| 61 | \( 1 - 572 T + p^{3} T^{2} \) |
| 67 | \( 1 + 108 T + p^{3} T^{2} \) |
| 71 | \( 1 + 560 T + p^{3} T^{2} \) |
| 73 | \( 1 - 586 T + p^{3} T^{2} \) |
| 79 | \( 1 + 57 T + p^{3} T^{2} \) |
| 83 | \( 1 - 252 T + p^{3} T^{2} \) |
| 89 | \( 1 + 184 T + p^{3} T^{2} \) |
| 97 | \( 1 + 605 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.674717883106783037203327617318, −7.987663778309495520685682299894, −7.13259179533408116686550454445, −6.11341870305509726975310505009, −5.68775176046337659482767658209, −4.80182917791303499328793010140, −3.80460204018647260096072829867, −2.49695375271611677216812103429, −2.20000527102684188666031873162, −0.21341250389132620517527842703,
0.21341250389132620517527842703, 2.20000527102684188666031873162, 2.49695375271611677216812103429, 3.80460204018647260096072829867, 4.80182917791303499328793010140, 5.68775176046337659482767658209, 6.11341870305509726975310505009, 7.13259179533408116686550454445, 7.987663778309495520685682299894, 8.674717883106783037203327617318
