| L(s) = 1 | − 2.18·2-s − 3.20·4-s − 11.9·5-s − 36.1·7-s + 24.5·8-s + 26.2·10-s − 21.1·11-s − 42.7·13-s + 79.1·14-s − 28.0·16-s + 104.·17-s + 3.53·19-s + 38.4·20-s + 46.3·22-s − 23·23-s + 18.4·25-s + 93.6·26-s + 115.·28-s − 56.5·29-s − 36.8·31-s − 134.·32-s − 228.·34-s + 433.·35-s + 45.4·37-s − 7.73·38-s − 293.·40-s + 458.·41-s + ⋯ |
| L(s) = 1 | − 0.774·2-s − 0.400·4-s − 1.07·5-s − 1.95·7-s + 1.08·8-s + 0.829·10-s − 0.580·11-s − 0.912·13-s + 1.51·14-s − 0.438·16-s + 1.48·17-s + 0.0426·19-s + 0.429·20-s + 0.449·22-s − 0.208·23-s + 0.147·25-s + 0.706·26-s + 0.782·28-s − 0.362·29-s − 0.213·31-s − 0.744·32-s − 1.15·34-s + 2.09·35-s + 0.202·37-s − 0.0330·38-s − 1.16·40-s + 1.74·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3107479816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3107479816\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 2.18T + 8T^{2} \) |
| 5 | \( 1 + 11.9T + 125T^{2} \) |
| 7 | \( 1 + 36.1T + 343T^{2} \) |
| 11 | \( 1 + 21.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 42.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 3.53T + 6.85e3T^{2} \) |
| 29 | \( 1 + 56.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 36.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 45.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 458.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 23.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 191.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 273.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 846.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 386.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 489.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 767.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 592.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 124.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 345.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−12.12513824376406343997274758529, −10.61592125473309098648767559665, −9.832385537502145665502570309069, −9.170675978941771209829660923473, −7.81971349524864697567369994924, −7.28887574292134761758444983104, −5.73206156287880404139187362270, −4.17231258515693335381530269346, −3.05203697766192185391647555891, −0.45350742923805704074649034012,
0.45350742923805704074649034012, 3.05203697766192185391647555891, 4.17231258515693335381530269346, 5.73206156287880404139187362270, 7.28887574292134761758444983104, 7.81971349524864697567369994924, 9.170675978941771209829660923473, 9.832385537502145665502570309069, 10.61592125473309098648767559665, 12.12513824376406343997274758529
