| L(s) = 1 | − 2-s + 4-s − 2·5-s − 7-s − 8-s + 2·10-s − 4·11-s − 13-s + 14-s + 16-s + 6·17-s − 2·20-s + 4·22-s − 8·23-s − 25-s + 26-s − 28-s + 10·29-s − 8·31-s − 32-s − 6·34-s + 2·35-s + 6·37-s + 2·40-s + 6·41-s + 4·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.447·20-s + 0.852·22-s − 1.66·23-s − 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.85·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.338·35-s + 0.986·37-s + 0.316·40-s + 0.937·41-s + 0.609·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7070504174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7070504174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.510038381479805846129615572317, −8.369588935348609600456733576104, −7.80088849136349478139956350243, −7.40803372522333751876994487224, −6.19483789697770666814563584573, −5.44024157926880829299881771792, −4.23885567109427787805557130576, −3.27331439299451669391457686232, −2.29694658439320588323627036273, −0.62603155646423681877598241271,
0.62603155646423681877598241271, 2.29694658439320588323627036273, 3.27331439299451669391457686232, 4.23885567109427787805557130576, 5.44024157926880829299881771792, 6.19483789697770666814563584573, 7.40803372522333751876994487224, 7.80088849136349478139956350243, 8.369588935348609600456733576104, 9.510038381479805846129615572317
