| L(s) = 1 | − 4.92·2-s + 16.2·4-s + 34.7·7-s − 40.7·8-s − 12.4·11-s + 37.1·13-s − 171.·14-s + 70.7·16-s + 102.·17-s + 63.7·19-s + 61.3·22-s + 100.·23-s − 182.·26-s + 566.·28-s + 138.·29-s + 24.8·31-s − 22.4·32-s − 505.·34-s + 186.·37-s − 314.·38-s − 255.·41-s − 108.·43-s − 202.·44-s − 495.·46-s − 223.·47-s + 866.·49-s + 604.·52-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 2.03·4-s + 1.87·7-s − 1.80·8-s − 0.341·11-s + 0.791·13-s − 3.27·14-s + 1.10·16-s + 1.46·17-s + 0.770·19-s + 0.594·22-s + 0.911·23-s − 1.37·26-s + 3.82·28-s + 0.889·29-s + 0.144·31-s − 0.123·32-s − 2.55·34-s + 0.829·37-s − 1.34·38-s − 0.971·41-s − 0.384·43-s − 0.694·44-s − 1.58·46-s − 0.694·47-s + 2.52·49-s + 1.61·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.305752351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.305752351\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 4.92T + 8T^{2} \) |
| 7 | \( 1 - 34.7T + 343T^{2} \) |
| 11 | \( 1 + 12.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 63.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 24.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 186.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 255.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 108.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 40.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 131.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 336.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 17.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 992.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 769.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 252.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 234.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 405.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.06e3T + 9.12e5T^{2} \) |
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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
−10.13203545039518592199515834194, −9.094105001181099121413226127903, −8.274655509972457354958061179982, −7.87934458002379504632438663329, −7.06756531813692825509573048869, −5.74808270151959549536907749657, −4.73586606669985809471519393401, −3.00876487163333010393709910409, −1.59581099895159972613977646973, −0.980669203994164410294914505573,
0.980669203994164410294914505573, 1.59581099895159972613977646973, 3.00876487163333010393709910409, 4.73586606669985809471519393401, 5.74808270151959549536907749657, 7.06756531813692825509573048869, 7.87934458002379504632438663329, 8.274655509972457354958061179982, 9.094105001181099121413226127903, 10.13203545039518592199515834194
