| L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s + 7-s + 3·8-s + 9-s − 10-s + 2·12-s + 13-s − 14-s − 2·15-s − 16-s − 3·17-s − 18-s + 4·19-s − 20-s − 2·21-s + 2·23-s − 6·24-s − 4·25-s − 26-s + 4·27-s − 28-s − 3·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s + 0.277·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.436·21-s + 0.417·23-s − 1.22·24-s − 4/5·25-s − 0.196·26-s + 0.769·27-s − 0.188·28-s − 0.557·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124579 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.71881957881876, −13.31880442566994, −12.86495282761977, −12.13165695800243, −11.68463970527457, −11.36649079416692, −10.81163169865780, −10.37960714119734, −9.860008561731577, −9.517076524986312, −8.934025421909846, −8.326689799565045, −8.050882965198634, −7.354996770746124, −6.765409084078222, −6.261475136754730, −5.788271729776862, −5.128163713342227, −4.776714535720735, −4.397753197736581, −3.457910939449005, −2.903498972361836, −1.799464545597874, −1.478588432182421, −0.6626750762901863, 0,
0.6626750762901863, 1.478588432182421, 1.799464545597874, 2.903498972361836, 3.457910939449005, 4.397753197736581, 4.776714535720735, 5.128163713342227, 5.788271729776862, 6.261475136754730, 6.765409084078222, 7.354996770746124, 8.050882965198634, 8.326689799565045, 8.934025421909846, 9.517076524986312, 9.860008561731577, 10.37960714119734, 10.81163169865780, 11.36649079416692, 11.68463970527457, 12.13165695800243, 12.86495282761977, 13.31880442566994, 13.71881957881876
