L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 3·11-s − 12-s − 2·13-s + 14-s + 16-s + 17-s − 18-s + 21-s − 3·22-s − 8·23-s + 24-s + 2·26-s − 27-s − 28-s + 8·29-s + 2·31-s − 32-s − 3·33-s − 34-s + 36-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 + 0.904·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.218·21-s − 0.639·22-s − 1.66·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.48·29-s + 0.359·31-s − 0.176·32-s − 0.522·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3174607055\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3174607055\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−12.20835880061570, −12.02099436423270, −11.71496312073335, −11.11863966525999, −10.45268545506885, −10.18882832804406, −9.952212278059488, −9.333443854040421, −8.858189807129864, −8.477893035084035, −7.893742551915822, −7.431277678261497, −6.936609057899964, −6.469252161778341, −6.179200285228621, −5.586615054776184, −5.114115219853070, −4.351237945703037, −4.076082074857946, −3.377151019913418, −2.783561442872357, −2.219091522081299, −1.486927684402323, −1.111029080693743, −0.1818533451635007,
0.1818533451635007, 1.111029080693743, 1.486927684402323, 2.219091522081299, 2.783561442872357, 3.377151019913418, 4.076082074857946, 4.351237945703037, 5.114115219853070, 5.586615054776184, 6.179200285228621, 6.469252161778341, 6.936609057899964, 7.431277678261497, 7.893742551915822, 8.477893035084035, 8.858189807129864, 9.333443854040421, 9.952212278059488, 10.18882832804406, 10.45268545506885, 11.11863966525999, 11.71496312073335, 12.02099436423270, 12.20835880061570