| L(s) = 1 | + 1.10·2-s − 1.12·3-s − 0.787·4-s − 1.24·6-s − 1.81·7-s − 3.06·8-s − 1.73·9-s + 11-s + 0.886·12-s − 3.00·13-s − 2.00·14-s − 1.80·16-s − 0.396·17-s − 1.90·18-s − 19-s + 2.04·21-s + 1.10·22-s − 2.47·23-s + 3.45·24-s − 3.31·26-s + 5.32·27-s + 1.43·28-s − 4.66·29-s − 10.5·31-s + 4.15·32-s − 1.12·33-s − 0.436·34-s + ⋯ |
| L(s) = 1 | + 0.778·2-s − 0.650·3-s − 0.393·4-s − 0.506·6-s − 0.686·7-s − 1.08·8-s − 0.577·9-s + 0.301·11-s + 0.255·12-s − 0.833·13-s − 0.534·14-s − 0.451·16-s − 0.0960·17-s − 0.449·18-s − 0.229·19-s + 0.446·21-s + 0.234·22-s − 0.515·23-s + 0.705·24-s − 0.649·26-s + 1.02·27-s + 0.270·28-s − 0.866·29-s − 1.90·31-s + 0.733·32-s − 0.196·33-s − 0.0748·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5914622649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5914622649\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 13 | \( 1 + 3.00T + 13T^{2} \) |
| 17 | \( 1 + 0.396T + 17T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 + 6.08T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 9.86T + 67T^{2} \) |
| 71 | \( 1 - 3.14T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 8.40T + 83T^{2} \) |
| 89 | \( 1 + 4.18T + 89T^{2} \) |
| 97 | \( 1 - 5.66T + 97T^{2} \) |
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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
−8.270147555644623033497339928341, −7.24507085639014016450589510222, −6.54741993249626951913549204647, −5.82968430959716861734769543515, −5.36335852840322909277633580492, −4.62740721902219108648148082920, −3.75385733425351102355655918063, −3.13468930779850516173926010666, −2.07134974930221152501732651857, −0.36054296666983698895810426076,
0.36054296666983698895810426076, 2.07134974930221152501732651857, 3.13468930779850516173926010666, 3.75385733425351102355655918063, 4.62740721902219108648148082920, 5.36335852840322909277633580492, 5.82968430959716861734769543515, 6.54741993249626951913549204647, 7.24507085639014016450589510222, 8.270147555644623033497339928341
