| L(s) = 1 | + 7.34·3-s − 15.7·5-s − 29.0·7-s + 26.8·9-s − 33.7·11-s − 13·13-s − 115.·15-s + 15.5·17-s + 103.·19-s − 213.·21-s + 145.·23-s + 122.·25-s − 0.906·27-s + 285.·29-s + 221.·31-s − 247.·33-s + 457.·35-s − 161.·37-s − 95.4·39-s − 404.·41-s + 198.·43-s − 422.·45-s − 125.·47-s + 501.·49-s + 114.·51-s + 548.·53-s + 530.·55-s + ⋯ |
| L(s) = 1 | + 1.41·3-s − 1.40·5-s − 1.56·7-s + 0.995·9-s − 0.924·11-s − 0.277·13-s − 1.98·15-s + 0.221·17-s + 1.24·19-s − 2.21·21-s + 1.31·23-s + 0.979·25-s − 0.00646·27-s + 1.82·29-s + 1.28·31-s − 1.30·33-s + 2.20·35-s − 0.719·37-s − 0.391·39-s − 1.54·41-s + 0.705·43-s − 1.40·45-s − 0.389·47-s + 1.46·49-s + 0.313·51-s + 1.42·53-s + 1.29·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.776691049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.776691049\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 - 7.34T + 27T^{2} \) |
| 5 | \( 1 + 15.7T + 125T^{2} \) |
| 7 | \( 1 + 29.0T + 343T^{2} \) |
| 11 | \( 1 + 33.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 15.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 285.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 221.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 404.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 125.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 548.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 536.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 643.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 171.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 480.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 353.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 263.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 458.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 831.T + 9.12e5T^{2} \) |
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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
−9.822245566989839763502960165663, −8.759912773992471447503292940106, −8.251321034720705417925410084001, −7.34189881984823842396622974372, −6.80273421501053683577543376132, −5.22837924627923732185391809415, −4.02168056986208182997104987042, −3.04977562129084836860345280634, −2.86619313230809329393704053635, −0.66704096019599527810324455387,
0.66704096019599527810324455387, 2.86619313230809329393704053635, 3.04977562129084836860345280634, 4.02168056986208182997104987042, 5.22837924627923732185391809415, 6.80273421501053683577543376132, 7.34189881984823842396622974372, 8.251321034720705417925410084001, 8.759912773992471447503292940106, 9.822245566989839763502960165663
