| L(s) = 1 | + 2·3-s + 16·5-s − 7·7-s − 23·9-s − 24·11-s + 68·13-s + 32·15-s + 54·17-s + 46·19-s − 14·21-s + 176·23-s + 131·25-s − 100·27-s + 174·29-s − 116·31-s − 48·33-s − 112·35-s − 74·37-s + 136·39-s − 10·41-s + 480·43-s − 368·45-s − 572·47-s + 49·49-s + 108·51-s + 162·53-s − 384·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.43·5-s − 0.377·7-s − 0.851·9-s − 0.657·11-s + 1.45·13-s + 0.550·15-s + 0.770·17-s + 0.555·19-s − 0.145·21-s + 1.59·23-s + 1.04·25-s − 0.712·27-s + 1.11·29-s − 0.672·31-s − 0.253·33-s − 0.540·35-s − 0.328·37-s + 0.558·39-s − 0.0380·41-s + 1.70·43-s − 1.21·45-s − 1.77·47-s + 1/7·49-s + 0.296·51-s + 0.419·53-s − 0.941·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.801768289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.801768289\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 46 T + p^{3} T^{2} \) |
| 23 | \( 1 - 176 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 116 T + p^{3} T^{2} \) |
| 37 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 10 T + p^{3} T^{2} \) |
| 43 | \( 1 - 480 T + p^{3} T^{2} \) |
| 47 | \( 1 + 572 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 86 T + p^{3} T^{2} \) |
| 61 | \( 1 - 904 T + p^{3} T^{2} \) |
| 67 | \( 1 + 660 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 73 | \( 1 - 770 T + p^{3} T^{2} \) |
| 79 | \( 1 + 904 T + p^{3} T^{2} \) |
| 83 | \( 1 + 682 T + p^{3} T^{2} \) |
| 89 | \( 1 + 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 218 T + p^{3} T^{2} \) |
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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
−10.59559560445141832767446428592, −9.693976854559557630387542086095, −8.959727651652422063315162854191, −8.160333572864296975887471983427, −6.81743636139393654954061326672, −5.84128449076005045911467724644, −5.25261638779245070783695689932, −3.41827121980011840626254517916, −2.55676654815546571740034622108, −1.12532286191305595436859421271,
1.12532286191305595436859421271, 2.55676654815546571740034622108, 3.41827121980011840626254517916, 5.25261638779245070783695689932, 5.84128449076005045911467724644, 6.81743636139393654954061326672, 8.160333572864296975887471983427, 8.959727651652422063315162854191, 9.693976854559557630387542086095, 10.59559560445141832767446428592
