| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 36·6-s − 49·7-s + 64·8-s + 81·9-s − 414·11-s − 144·12-s + 1.05e3·13-s − 196·14-s + 256·16-s + 1.84e3·17-s + 324·18-s + 236·19-s + 441·21-s − 1.65e3·22-s − 2.89e3·23-s − 576·24-s + 4.21e3·26-s − 729·27-s − 784·28-s − 6.52e3·29-s + 6.20e3·31-s + 1.02e3·32-s + 3.72e3·33-s + 7.39e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.03·11-s − 0.288·12-s + 1.72·13-s − 0.267·14-s + 1/4·16-s + 1.55·17-s + 0.235·18-s + 0.149·19-s + 0.218·21-s − 0.729·22-s − 1.14·23-s − 0.204·24-s + 1.22·26-s − 0.192·27-s − 0.188·28-s − 1.44·29-s + 1.15·31-s + 0.176·32-s + 0.595·33-s + 1.09·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.874456921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.874456921\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 11 | \( 1 + 414 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1054 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1848 T + p^{5} T^{2} \) |
| 19 | \( 1 - 236 T + p^{5} T^{2} \) |
| 23 | \( 1 + 126 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 6522 T + p^{5} T^{2} \) |
| 31 | \( 1 - 200 p T + p^{5} T^{2} \) |
| 37 | \( 1 + 9650 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8484 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10804 T + p^{5} T^{2} \) |
| 47 | \( 1 + 60 T + p^{5} T^{2} \) |
| 53 | \( 1 + 22506 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28176 T + p^{5} T^{2} \) |
| 61 | \( 1 + 35194 T + p^{5} T^{2} \) |
| 67 | \( 1 - 28216 T + p^{5} T^{2} \) |
| 71 | \( 1 + 6642 T + p^{5} T^{2} \) |
| 73 | \( 1 - 52090 T + p^{5} T^{2} \) |
| 79 | \( 1 - 43340 T + p^{5} T^{2} \) |
| 83 | \( 1 + 25716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 98724 T + p^{5} T^{2} \) |
| 97 | \( 1 - 148954 T + p^{5} T^{2} \) |
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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
−9.312971999618925928274754709805, −8.069528213199890964650618271699, −7.52142105401098834192547174571, −6.21781225315562233248133613745, −5.88072604437491826583097388717, −4.99895931389882960213235902524, −3.84255628003179066151550620319, −3.17129392329209085282468751558, −1.80511020565058431550514173716, −0.68467325073577799935899509271,
0.68467325073577799935899509271, 1.80511020565058431550514173716, 3.17129392329209085282468751558, 3.84255628003179066151550620319, 4.99895931389882960213235902524, 5.88072604437491826583097388717, 6.21781225315562233248133613745, 7.52142105401098834192547174571, 8.069528213199890964650618271699, 9.312971999618925928274754709805
