L(s) = 1 | + 3.10·3-s + 2.81·5-s + 7-s + 6.62·9-s − 3.10·11-s + 13-s + 8.72·15-s − 0.524·17-s − 0.813·19-s + 3.10·21-s − 7.33·23-s + 2.91·25-s + 11.2·27-s + 8.28·29-s − 1.39·31-s − 9.62·33-s + 2.81·35-s − 6.15·37-s + 3.10·39-s − 4.20·41-s − 6.75·43-s + 18.6·45-s + 5.97·47-s + 49-s − 1.62·51-s − 2.49·53-s − 8.72·55-s + ⋯ |
L(s) = 1 | + 1.79·3-s + 1.25·5-s + 0.377·7-s + 2.20·9-s − 0.935·11-s + 0.277·13-s + 2.25·15-s − 0.127·17-s − 0.186·19-s + 0.677·21-s − 1.53·23-s + 0.583·25-s + 2.16·27-s + 1.53·29-s − 0.250·31-s − 1.67·33-s + 0.475·35-s − 1.01·37-s + 0.496·39-s − 0.656·41-s − 1.03·43-s + 2.77·45-s + 0.870·47-s + 0.142·49-s − 0.227·51-s − 0.342·53-s − 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.904448448\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.904448448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 17 | \( 1 + 0.524T + 17T^{2} \) |
| 19 | \( 1 + 0.813T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 - 8.28T + 29T^{2} \) |
| 31 | \( 1 + 1.39T + 31T^{2} \) |
| 37 | \( 1 + 6.15T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 + 6.75T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 + 2.49T + 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 8.72T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.18T + 97T^{2} \) |
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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
−9.495481766396675463960404906458, −8.547364917649146820897717181080, −8.243895731884702410457988735007, −7.29783290257583953046814290682, −6.34009340493963960645564380591, −5.30493396736537125383925114915, −4.29190655729097811687734341500, −3.19442215475345536716127725010, −2.31185673482776118411148128145, −1.67601087289506257611219709057,
1.67601087289506257611219709057, 2.31185673482776118411148128145, 3.19442215475345536716127725010, 4.29190655729097811687734341500, 5.30493396736537125383925114915, 6.34009340493963960645564380591, 7.29783290257583953046814290682, 8.243895731884702410457988735007, 8.547364917649146820897717181080, 9.495481766396675463960404906458