L(s) = 1 | + 1.42·2-s − 9·3-s − 29.9·4-s − 83.8·5-s − 12.8·6-s + 175.·7-s − 88.5·8-s + 81·9-s − 119.·10-s + 373.·11-s + 269.·12-s + 930.·13-s + 250.·14-s + 755.·15-s + 832.·16-s − 1.10e3·17-s + 115.·18-s − 2.42e3·19-s + 2.51e3·20-s − 1.58e3·21-s + 533.·22-s + 529·23-s + 796.·24-s + 3.91e3·25-s + 1.33e3·26-s − 729·27-s − 5.26e3·28-s + ⋯ |
L(s) = 1 | + 0.252·2-s − 0.577·3-s − 0.936·4-s − 1.50·5-s − 0.145·6-s + 1.35·7-s − 0.489·8-s + 0.333·9-s − 0.379·10-s + 0.930·11-s + 0.540·12-s + 1.52·13-s + 0.342·14-s + 0.866·15-s + 0.812·16-s − 0.930·17-s + 0.0841·18-s − 1.54·19-s + 1.40·20-s − 0.782·21-s + 0.234·22-s + 0.208·23-s + 0.282·24-s + 1.25·25-s + 0.385·26-s − 0.192·27-s − 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.124387034\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.124387034\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
good | 2 | \( 1 - 1.42T + 32T^{2} \) |
| 5 | \( 1 + 83.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 175.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 373.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 930.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.10e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.42e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 7.81e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.72e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.22e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.38e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.00e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.95e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.24e4T + 8.58e9T^{2} \) |
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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
−13.78733439981809371876831295263, −12.48932278333697631523503824702, −11.51090618669035492196798504440, −10.82383021228841810380783455552, −8.733366182979659215949909794024, −8.157660447568575287993811111533, −6.39480949982109111617830612074, −4.57649430467843093323078164884, −4.04508172993792530991655319970, −0.871856112614604151453973578648,
0.871856112614604151453973578648, 4.04508172993792530991655319970, 4.57649430467843093323078164884, 6.39480949982109111617830612074, 8.157660447568575287993811111533, 8.733366182979659215949909794024, 10.82383021228841810380783455552, 11.51090618669035492196798504440, 12.48932278333697631523503824702, 13.78733439981809371876831295263