| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 10·5-s − 6·6-s − 8·7-s − 8·8-s + 9·9-s − 20·10-s + 40·11-s + 12·12-s + 13·13-s + 16·14-s + 30·15-s + 16·16-s + 130·17-s − 18·18-s − 20·19-s + 40·20-s − 24·21-s − 80·22-s − 24·24-s − 25·25-s − 26·26-s + 27·27-s − 32·28-s − 18·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.431·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.09·11-s + 0.288·12-s + 0.277·13-s + 0.305·14-s + 0.516·15-s + 1/4·16-s + 1.85·17-s − 0.235·18-s − 0.241·19-s + 0.447·20-s − 0.249·21-s − 0.775·22-s − 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.215·28-s − 0.115·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.518843172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.518843172\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 17 | \( 1 - 130 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 18 T + p^{3} T^{2} \) |
| 31 | \( 1 + 184 T + p^{3} T^{2} \) |
| 37 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 362 T + p^{3} T^{2} \) |
| 43 | \( 1 - 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 452 T + p^{3} T^{2} \) |
| 53 | \( 1 - 382 T + p^{3} T^{2} \) |
| 59 | \( 1 - 464 T + p^{3} T^{2} \) |
| 61 | \( 1 - 358 T + p^{3} T^{2} \) |
| 67 | \( 1 + 700 T + p^{3} T^{2} \) |
| 71 | \( 1 + 748 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1058 T + p^{3} T^{2} \) |
| 79 | \( 1 + 976 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1008 T + p^{3} T^{2} \) |
| 89 | \( 1 + 386 T + p^{3} T^{2} \) |
| 97 | \( 1 + 614 T + p^{3} T^{2} \) |
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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
−14.09756807789829120084423529219, −12.91570494528019113435940065811, −11.71138578085443686332141895308, −10.14962994950704125001025271428, −9.533349560254652641545745262612, −8.442498337872731240943690035565, −7.03033249388355308990852937499, −5.78100985038299022192291316608, −3.43742783075325042177745966542, −1.58989444307572857990467265711,
1.58989444307572857990467265711, 3.43742783075325042177745966542, 5.78100985038299022192291316608, 7.03033249388355308990852937499, 8.442498337872731240943690035565, 9.533349560254652641545745262612, 10.14962994950704125001025271428, 11.71138578085443686332141895308, 12.91570494528019113435940065811, 14.09756807789829120084423529219
