| L(s) = 1 | − 2.01·2-s − 3.95·4-s + 6.74·5-s + 7.15·7-s + 24.0·8-s − 13.5·10-s − 55.3·11-s − 49.0·13-s − 14.3·14-s − 16.7·16-s − 107.·17-s − 146.·19-s − 26.6·20-s + 111.·22-s + 154.·23-s − 79.5·25-s + 98.7·26-s − 28.2·28-s − 211.·29-s − 128.·31-s − 158.·32-s + 217.·34-s + 48.2·35-s − 106.·37-s + 294.·38-s + 162.·40-s + 245.·41-s + ⋯ |
| L(s) = 1 | − 0.711·2-s − 0.494·4-s + 0.602·5-s + 0.386·7-s + 1.06·8-s − 0.428·10-s − 1.51·11-s − 1.04·13-s − 0.274·14-s − 0.261·16-s − 1.53·17-s − 1.76·19-s − 0.298·20-s + 1.07·22-s + 1.40·23-s − 0.636·25-s + 0.744·26-s − 0.190·28-s − 1.35·29-s − 0.741·31-s − 0.876·32-s + 1.09·34-s + 0.232·35-s − 0.471·37-s + 1.25·38-s + 0.640·40-s + 0.933·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3530411390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3530411390\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 2.01T + 8T^{2} \) |
| 5 | \( 1 - 6.74T + 125T^{2} \) |
| 7 | \( 1 - 7.15T + 343T^{2} \) |
| 11 | \( 1 + 55.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 211.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 106.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 245.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 330.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 146.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 245.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 650.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 121.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 126.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 773.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 607.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 119.T + 9.12e5T^{2} \) |
| 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
−8.786153705174488007106077823990, −8.083521532073994687853664462121, −7.37664680323561279740576677536, −6.54753073013665307080106561397, −5.27438932839460523052580049613, −4.93844316639714880907389806830, −3.94903947058232794860329378396, −2.39420604195204663280501025982, −1.91504850734282737342035370940, −0.28559759605906349214375982714,
0.28559759605906349214375982714, 1.91504850734282737342035370940, 2.39420604195204663280501025982, 3.94903947058232794860329378396, 4.93844316639714880907389806830, 5.27438932839460523052580049613, 6.54753073013665307080106561397, 7.37664680323561279740576677536, 8.083521532073994687853664462121, 8.786153705174488007106077823990
