| L(s) = 1 | − 3·3-s + 5·5-s + 28·7-s + 9·9-s − 24·11-s + 70·13-s − 15·15-s + 102·17-s + 20·19-s − 84·21-s + 72·23-s + 25·25-s − 27·27-s − 306·29-s + 136·31-s + 72·33-s + 140·35-s + 214·37-s − 210·39-s − 150·41-s − 292·43-s + 45·45-s + 72·47-s + 441·49-s − 306·51-s + 414·53-s − 120·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.657·11-s + 1.49·13-s − 0.258·15-s + 1.45·17-s + 0.241·19-s − 0.872·21-s + 0.652·23-s + 1/5·25-s − 0.192·27-s − 1.95·29-s + 0.787·31-s + 0.379·33-s + 0.676·35-s + 0.950·37-s − 0.862·39-s − 0.571·41-s − 1.03·43-s + 0.149·45-s + 0.223·47-s + 9/7·49-s − 0.840·51-s + 1.07·53-s − 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.586077241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.586077241\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 306 T + p^{3} T^{2} \) |
| 31 | \( 1 - 136 T + p^{3} T^{2} \) |
| 37 | \( 1 - 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 150 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 418 T + p^{3} T^{2} \) |
| 67 | \( 1 - 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 480 T + p^{3} T^{2} \) |
| 73 | \( 1 - 434 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1352 T + p^{3} T^{2} \) |
| 83 | \( 1 + 612 T + p^{3} T^{2} \) |
| 89 | \( 1 + 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−9.795576726134708194380337698980, −8.682435053424911872079024591970, −7.976884651422769913868263233472, −7.20170343182345390506477755756, −5.88057793122348246905630066176, −5.45435193794448364281267718353, −4.52149551638428517826763589465, −3.31420653071850353559455448901, −1.79033795035537152064945323054, −0.982464350390668153798959594149,
0.982464350390668153798959594149, 1.79033795035537152064945323054, 3.31420653071850353559455448901, 4.52149551638428517826763589465, 5.45435193794448364281267718353, 5.88057793122348246905630066176, 7.20170343182345390506477755756, 7.976884651422769913868263233472, 8.682435053424911872079024591970, 9.795576726134708194380337698980
