| L(s) = 1 | + 1.71·3-s − 2.77·7-s − 0.0597·9-s − 2.77·11-s − 5.67·13-s − 5.15·17-s − 1.41·19-s − 4.75·21-s − 0.654·23-s − 5.24·27-s + 4.09·29-s + 7.12·31-s − 4.75·33-s + 1.04·37-s − 9.73·39-s + 9.10·41-s + 9.24·43-s − 2.77·47-s + 0.697·49-s − 8.84·51-s + 0.526·53-s − 2.42·57-s + 3.78·59-s − 10.8·61-s + 0.165·63-s − 4.32·67-s − 1.12·69-s + ⋯ |
| L(s) = 1 | + 0.989·3-s − 1.04·7-s − 0.0199·9-s − 0.836·11-s − 1.57·13-s − 1.25·17-s − 0.324·19-s − 1.03·21-s − 0.136·23-s − 1.00·27-s + 0.760·29-s + 1.28·31-s − 0.828·33-s + 0.172·37-s − 1.55·39-s + 1.42·41-s + 1.41·43-s − 0.404·47-s + 0.0996·49-s − 1.23·51-s + 0.0722·53-s − 0.320·57-s + 0.492·59-s − 1.38·61-s + 0.0208·63-s − 0.528·67-s − 0.135·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.289474704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.289474704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.71T + 3T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 + 2.77T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 + 0.654T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 - 1.04T + 37T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 + 2.77T + 47T^{2} \) |
| 53 | \( 1 - 0.526T + 53T^{2} \) |
| 59 | \( 1 - 3.78T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 4.32T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 4.21T + 73T^{2} \) |
| 79 | \( 1 - 9.90T + 79T^{2} \) |
| 83 | \( 1 - 4.67T + 83T^{2} \) |
| 89 | \( 1 + 9.18T + 89T^{2} \) |
| 97 | \( 1 - 0.0901T + 97T^{2} \) |
| 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
−7.80788734025687144760987542067, −7.03008211202687654800764095739, −6.41338539944854944771195946816, −5.67511314248413094469286023701, −4.71983446957965921618172746142, −4.19476388804737672948515038172, −3.10962285002614148066206569058, −2.61910747585968616596917198427, −2.19303675628893038256483412883, −0.46684339968310399338868529151,
0.46684339968310399338868529151, 2.19303675628893038256483412883, 2.61910747585968616596917198427, 3.10962285002614148066206569058, 4.19476388804737672948515038172, 4.71983446957965921618172746142, 5.67511314248413094469286023701, 6.41338539944854944771195946816, 7.03008211202687654800764095739, 7.80788734025687144760987542067
