| L(s) = 1 | − 5.19·2-s − 3·3-s + 19.0·4-s − 8.30·5-s + 15.5·6-s + 21.5·7-s − 57.1·8-s + 9·9-s + 43.1·10-s − 28.3·11-s − 57.0·12-s − 28.2·13-s − 111.·14-s + 24.9·15-s + 145.·16-s + 21.8·17-s − 46.7·18-s − 122.·19-s − 157.·20-s − 64.6·21-s + 147.·22-s + 82.1·23-s + 171.·24-s − 56.0·25-s + 146.·26-s − 27·27-s + 409.·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.37·4-s − 0.742·5-s + 1.06·6-s + 1.16·7-s − 2.52·8-s + 0.333·9-s + 1.36·10-s − 0.777·11-s − 1.37·12-s − 0.601·13-s − 2.13·14-s + 0.428·15-s + 2.26·16-s + 0.311·17-s − 0.612·18-s − 1.47·19-s − 1.76·20-s − 0.671·21-s + 1.42·22-s + 0.744·23-s + 1.45·24-s − 0.448·25-s + 1.10·26-s − 0.192·27-s + 2.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4615016593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4615016593\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
| good | 2 | \( 1 + 5.19T + 8T^{2} \) |
| 5 | \( 1 + 8.30T + 125T^{2} \) |
| 7 | \( 1 - 21.5T + 343T^{2} \) |
| 11 | \( 1 + 28.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 21.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 82.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 86.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 381.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 452.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 158.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 162.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 368.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 177.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 58.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 880.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 825.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.42e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.55e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.81e3T + 9.12e5T^{2} \) |
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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
−11.69758965620666144053040857389, −11.00082932430058605846862264539, −10.36847204527730390538517454634, −9.136720095602382159488128778291, −7.943345921992740838584923335550, −7.66358134356174094262280935708, −6.27005569144307146823893493993, −4.65582734937962653102870568504, −2.32616077856440584250832541071, −0.70692408628685967928988057097,
0.70692408628685967928988057097, 2.32616077856440584250832541071, 4.65582734937962653102870568504, 6.27005569144307146823893493993, 7.66358134356174094262280935708, 7.943345921992740838584923335550, 9.136720095602382159488128778291, 10.36847204527730390538517454634, 11.00082932430058605846862264539, 11.69758965620666144053040857389
