L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 4·11-s + 12-s − 2·13-s + 15-s + 16-s − 6·17-s + 18-s + 4·19-s + 20-s + 4·22-s − 23-s + 24-s + 25-s − 2·26-s + 27-s − 2·29-s + 30-s + 32-s + 4·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.371·29-s + 0.182·30-s + 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.043931106\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.043931106\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.50540226217556192995644348238, −9.450916859891260810914138818635, −8.958168635991184817325388145038, −7.69536834975899346194495089447, −6.83892769208866057920177586677, −6.04043014103594893292288227968, −4.83247841860333266337206540504, −3.96718444193255144028903410718, −2.81374528766889432720312022940, −1.67150792184768704605428843936,
1.67150792184768704605428843936, 2.81374528766889432720312022940, 3.96718444193255144028903410718, 4.83247841860333266337206540504, 6.04043014103594893292288227968, 6.83892769208866057920177586677, 7.69536834975899346194495089447, 8.958168635991184817325388145038, 9.450916859891260810914138818635, 10.50540226217556192995644348238