| L(s) = 1 | − 0.649·3-s − 5-s + 1.74·7-s − 2.57·9-s − 6.54·11-s − 6.26·13-s + 0.649·15-s + 2.27·17-s − 7.79·19-s − 1.13·21-s − 4.43·23-s + 25-s + 3.62·27-s − 29-s + 0.487·31-s + 4.25·33-s − 1.74·35-s − 4.85·37-s + 4.06·39-s − 10.7·41-s − 5.57·43-s + 2.57·45-s − 2.71·47-s − 3.96·49-s − 1.48·51-s + 7.64·53-s + 6.54·55-s + ⋯ |
| L(s) = 1 | − 0.375·3-s − 0.447·5-s + 0.658·7-s − 0.859·9-s − 1.97·11-s − 1.73·13-s + 0.167·15-s + 0.552·17-s − 1.78·19-s − 0.247·21-s − 0.924·23-s + 0.200·25-s + 0.697·27-s − 0.185·29-s + 0.0876·31-s + 0.739·33-s − 0.294·35-s − 0.797·37-s + 0.651·39-s − 1.68·41-s − 0.849·43-s + 0.384·45-s − 0.395·47-s − 0.565·49-s − 0.207·51-s + 1.05·53-s + 0.882·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02848703362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02848703362\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 0.649T + 3T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 + 6.54T + 11T^{2} \) |
| 13 | \( 1 + 6.26T + 13T^{2} \) |
| 17 | \( 1 - 2.27T + 17T^{2} \) |
| 19 | \( 1 + 7.79T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 31 | \( 1 - 0.487T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 5.57T + 43T^{2} \) |
| 47 | \( 1 + 2.71T + 47T^{2} \) |
| 53 | \( 1 - 7.64T + 53T^{2} \) |
| 59 | \( 1 - 2.10T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 - 6.49T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 3.34T + 73T^{2} \) |
| 79 | \( 1 + 3.17T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 6.52T + 89T^{2} \) |
| 97 | \( 1 - 10.5T + 97T^{2} \) |
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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
−7.79063013420796151686565234403, −7.16380010600799707350732094488, −6.31268804875198334038866273145, −5.41240761134143320585999266784, −5.05592722025972861411360457570, −4.47901986350002166101190463927, −3.34176929046881741946233730129, −2.53003603690504068189591373477, −1.94675217427986356222371092557, −0.07549677052465045322288474123,
0.07549677052465045322288474123, 1.94675217427986356222371092557, 2.53003603690504068189591373477, 3.34176929046881741946233730129, 4.47901986350002166101190463927, 5.05592722025972861411360457570, 5.41240761134143320585999266784, 6.31268804875198334038866273145, 7.16380010600799707350732094488, 7.79063013420796151686565234403
