| L(s) = 1 | − 0.918·2-s − 7.15·4-s − 19.7·5-s + 10.7·7-s + 13.9·8-s + 18.0·10-s − 9.70·11-s + 45.1·13-s − 9.84·14-s + 44.4·16-s + 7.90·17-s + 6.45·19-s + 141.·20-s + 8.91·22-s + 192.·23-s + 263.·25-s − 41.4·26-s − 76.6·28-s − 188.·29-s + 176.·31-s − 152.·32-s − 7.25·34-s − 211.·35-s − 119.·37-s − 5.92·38-s − 274.·40-s − 201.·41-s + ⋯ |
| L(s) = 1 | − 0.324·2-s − 0.894·4-s − 1.76·5-s + 0.578·7-s + 0.615·8-s + 0.572·10-s − 0.266·11-s + 0.962·13-s − 0.187·14-s + 0.694·16-s + 0.112·17-s + 0.0779·19-s + 1.57·20-s + 0.0863·22-s + 1.74·23-s + 2.10·25-s − 0.312·26-s − 0.517·28-s − 1.20·29-s + 1.02·31-s − 0.840·32-s − 0.0366·34-s − 1.01·35-s − 0.530·37-s − 0.0252·38-s − 1.08·40-s − 0.767·41-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
| \(L(2)\) |
\(\approx\) |
\(0.8300142304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8300142304\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.918T + 8T^{2} \) |
| 5 | \( 1 + 19.7T + 125T^{2} \) |
| 7 | \( 1 - 10.7T + 343T^{2} \) |
| 11 | \( 1 + 9.70T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.90T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.45T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 119.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 201.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 402.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 446.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 470.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 667.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 564.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 737.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 539.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 803.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 21.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 417.T + 9.12e5T^{2} \) |
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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
−8.530559358337419299396858785447, −8.118874899757505068602064711323, −7.47073735198598577662452760068, −6.60927018677932381593740175112, −5.15272453773973349111001002100, −4.75629360651846857459314635371, −3.75081875421580997257972996855, −3.25474467379468009784017601739, −1.41295384913641833122399492307, −0.48334514411029409686000658002,
0.48334514411029409686000658002, 1.41295384913641833122399492307, 3.25474467379468009784017601739, 3.75081875421580997257972996855, 4.75629360651846857459314635371, 5.15272453773973349111001002100, 6.60927018677932381593740175112, 7.47073735198598577662452760068, 8.118874899757505068602064711323, 8.530559358337419299396858785447
