| L(s) = 1 | − 3.27·3-s + 6.27·5-s − 18.0·7-s − 16.2·9-s + 33.9·11-s + 3.07·13-s − 20.5·15-s − 14.2·17-s + 19·19-s + 59.2·21-s + 114.·23-s − 85.6·25-s + 141.·27-s + 34.5·29-s + 107.·31-s − 111.·33-s − 113.·35-s + 181.·37-s − 10.0·39-s − 444.·41-s − 120.·43-s − 102.·45-s + 306.·47-s − 15.4·49-s + 46.6·51-s − 115.·53-s + 212.·55-s + ⋯ |
| L(s) = 1 | − 0.630·3-s + 0.561·5-s − 0.977·7-s − 0.602·9-s + 0.929·11-s + 0.0656·13-s − 0.353·15-s − 0.203·17-s + 0.229·19-s + 0.615·21-s + 1.03·23-s − 0.685·25-s + 1.01·27-s + 0.221·29-s + 0.623·31-s − 0.586·33-s − 0.548·35-s + 0.807·37-s − 0.0413·39-s − 1.69·41-s − 0.426·43-s − 0.338·45-s + 0.950·47-s − 0.0449·49-s + 0.128·51-s − 0.300·53-s + 0.521·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 3.27T + 27T^{2} \) |
| 5 | \( 1 - 6.27T + 125T^{2} \) |
| 7 | \( 1 + 18.0T + 343T^{2} \) |
| 11 | \( 1 - 33.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.07T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.2T + 4.91e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 34.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 107.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 120.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 115.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 161.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 81.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 773.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 148.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 557.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 768.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 457.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.50e3T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.133331313173955218099166307665, −8.258131536973916879195364310255, −6.95027798524159053477813656577, −6.39044751246560997850331593987, −5.73853543167520030943981038393, −4.79278121740365533027018061276, −3.58395896478627954611022073176, −2.65557616798797870971833433627, −1.23161528137329077263778578881, 0,
1.23161528137329077263778578881, 2.65557616798797870971833433627, 3.58395896478627954611022073176, 4.79278121740365533027018061276, 5.73853543167520030943981038393, 6.39044751246560997850331593987, 6.95027798524159053477813656577, 8.258131536973916879195364310255, 9.133331313173955218099166307665
