| L(s) = 1 | − 3.57·2-s + 4.77·4-s + 5·5-s + 14.1·7-s + 11.5·8-s − 17.8·10-s + 63.6·11-s − 10.3·13-s − 50.6·14-s − 79.3·16-s + 108.·17-s + 18.6·19-s + 23.8·20-s − 227.·22-s − 128.·23-s + 25·25-s + 37.1·26-s + 67.6·28-s + 83.6·29-s − 10.4·31-s + 191.·32-s − 388.·34-s + 70.7·35-s − 81.8·37-s − 66.5·38-s + 57.5·40-s − 307.·41-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.597·4-s + 0.447·5-s + 0.764·7-s + 0.509·8-s − 0.565·10-s + 1.74·11-s − 0.221·13-s − 0.966·14-s − 1.24·16-s + 1.54·17-s + 0.224·19-s + 0.267·20-s − 2.20·22-s − 1.16·23-s + 0.200·25-s + 0.280·26-s + 0.456·28-s + 0.535·29-s − 0.0606·31-s + 1.05·32-s − 1.95·34-s + 0.341·35-s − 0.363·37-s − 0.284·38-s + 0.227·40-s − 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.308481393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308481393\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 3.57T + 8T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 11 | \( 1 - 63.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 18.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 128.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 83.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 10.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 81.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 222.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 361.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 562.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 462.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 649.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 254.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 524.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 656.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−10.46152494535476487391081216183, −9.852421151811940374971491294850, −9.039894387278464859861045469132, −8.247604152463826293555690077986, −7.37699835697880362784386720377, −6.33257901866337784078506624308, −5.05390515040547314639925860532, −3.77786686954606568848211216658, −1.86818162134862014644348393597, −0.991337785828049597545429245733,
0.991337785828049597545429245733, 1.86818162134862014644348393597, 3.77786686954606568848211216658, 5.05390515040547314639925860532, 6.33257901866337784078506624308, 7.37699835697880362784386720377, 8.247604152463826293555690077986, 9.039894387278464859861045469132, 9.852421151811940374971491294850, 10.46152494535476487391081216183
