| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 13-s + 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 4·22-s − 8·23-s + 24-s + 25-s − 26-s + 27-s − 6·29-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 − 0.277·13-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 − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 1.11·29-s + 0.182·30-s + 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.80270166420487, −12.23895431640180, −12.03611652852201, −11.43524159856915, −10.95521405947553, −10.24547836168597, −10.12990212119446, −9.551897390056662, −9.357428659421501, −8.369053929409002, −8.033051564047199, −7.744691109028412, −7.296714921293047, −6.701752015649362, −6.064246832998555, −5.677131948968659, −5.194736884697062, −4.928166545283525, −4.090181022509631, −3.646653604226630, −3.219314514803355, −2.573375716599178, −2.239280994567302, −1.597701581237092, −0.9384874727147908, 0,
0.9384874727147908, 1.597701581237092, 2.239280994567302, 2.573375716599178, 3.219314514803355, 3.646653604226630, 4.090181022509631, 4.928166545283525, 5.194736884697062, 5.677131948968659, 6.064246832998555, 6.701752015649362, 7.296714921293047, 7.744691109028412, 8.033051564047199, 8.369053929409002, 9.357428659421501, 9.551897390056662, 10.12990212119446, 10.24547836168597, 10.95521405947553, 11.43524159856915, 12.03611652852201, 12.23895431640180, 12.80270166420487
