| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s + 12-s + 4·13-s − 2·14-s − 15-s + 16-s + 6·17-s − 18-s − 2·19-s − 20-s + 2·21-s + 4·23-s − 24-s + 25-s − 4·26-s + 27-s + 2·28-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 + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.10·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.436·21-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.377·28-s + 0.371·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.945518721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.945518721\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.432401282520042939708637495904, −7.945296968192124290007262800115, −7.40886444281013931863745938030, −6.48290477175586103784418133268, −5.62784638242723502507010483765, −4.66631188689769848858236987716, −3.69199414920161283124226208743, −3.00120021488719719570320167565, −1.79012534218631982490471396889, −0.949444984762890882801218241897,
0.949444984762890882801218241897, 1.79012534218631982490471396889, 3.00120021488719719570320167565, 3.69199414920161283124226208743, 4.66631188689769848858236987716, 5.62784638242723502507010483765, 6.48290477175586103784418133268, 7.40886444281013931863745938030, 7.945296968192124290007262800115, 8.432401282520042939708637495904
