| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s − 7-s + 3·8-s + 9-s − 10-s − 12-s + 6·13-s + 14-s + 15-s − 16-s − 2·17-s − 18-s + 8·19-s − 20-s − 21-s + 8·23-s + 3·24-s + 25-s − 6·26-s + 27-s + 28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.218·21-s + 1.66·23-s + 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12705 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12705 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.293258721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293258721\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.43381976102608, −15.64562294117982, −15.44952416623586, −14.39998305881301, −13.95457875616437, −13.40267994392224, −13.22819455142484, −12.48472381360177, −11.51369522515386, −11.04599431029798, −10.25285825468811, −9.922237326238898, −9.105644864437313, −8.851022619880387, −8.397380255210656, −7.457939062160064, −7.084412241951065, −6.200102152928528, −5.467673994828118, −4.779995519486840, −3.919374968505393, −3.299603863549442, −2.507745424603696, −1.268470302593480, −0.9115067649991902,
0.9115067649991902, 1.268470302593480, 2.507745424603696, 3.299603863549442, 3.919374968505393, 4.779995519486840, 5.467673994828118, 6.200102152928528, 7.084412241951065, 7.457939062160064, 8.397380255210656, 8.851022619880387, 9.105644864437313, 9.922237326238898, 10.25285825468811, 11.04599431029798, 11.51369522515386, 12.48472381360177, 13.22819455142484, 13.40267994392224, 13.95457875616437, 14.39998305881301, 15.44952416623586, 15.64562294117982, 16.43381976102608
