| L(s) = 1 | + 2-s + 3·3-s − 7·4-s + 5·5-s + 3·6-s − 24·7-s − 15·8-s + 9·9-s + 5·10-s + 52·11-s − 21·12-s + 22·13-s − 24·14-s + 15·15-s + 41·16-s − 14·17-s + 9·18-s − 20·19-s − 35·20-s − 72·21-s + 52·22-s − 168·23-s − 45·24-s + 25·25-s + 22·26-s + 27·27-s + 168·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.577·3-s − 7/8·4-s + 0.447·5-s + 0.204·6-s − 1.29·7-s − 0.662·8-s + 1/3·9-s + 0.158·10-s + 1.42·11-s − 0.505·12-s + 0.469·13-s − 0.458·14-s + 0.258·15-s + 0.640·16-s − 0.199·17-s + 0.117·18-s − 0.241·19-s − 0.391·20-s − 0.748·21-s + 0.503·22-s − 1.52·23-s − 0.382·24-s + 1/5·25-s + 0.165·26-s + 0.192·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.139433421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.139433421\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 230 T + p^{3} T^{2} \) |
| 31 | \( 1 + 288 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 122 T + p^{3} T^{2} \) |
| 43 | \( 1 + 188 T + p^{3} T^{2} \) |
| 47 | \( 1 - 256 T + p^{3} T^{2} \) |
| 53 | \( 1 + 338 T + p^{3} T^{2} \) |
| 59 | \( 1 - 100 T + p^{3} T^{2} \) |
| 61 | \( 1 - 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 84 T + p^{3} T^{2} \) |
| 71 | \( 1 + 328 T + p^{3} T^{2} \) |
| 73 | \( 1 + 38 T + p^{3} T^{2} \) |
| 79 | \( 1 + 240 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 - 330 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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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
−19.02881405917150528687290867331, −17.70949836886805470548052216760, −16.19372858254051139851659443609, −14.50847812106069670742108136410, −13.56121835041038960562334260750, −12.39876426593880772586471717594, −9.853422404707595692052539772777, −8.847113744022096962511239249312, −6.30445355770960208718872883735, −3.78175129911665234242031093429,
3.78175129911665234242031093429, 6.30445355770960208718872883735, 8.847113744022096962511239249312, 9.853422404707595692052539772777, 12.39876426593880772586471717594, 13.56121835041038960562334260750, 14.50847812106069670742108136410, 16.19372858254051139851659443609, 17.70949836886805470548052216760, 19.02881405917150528687290867331
