| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 2·7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 13-s − 2·14-s + 15-s + 16-s − 8·17-s + 18-s − 3·19-s − 20-s + 2·21-s − 22-s − 9·23-s − 24-s + 25-s − 26-s − 27-s − 2·28-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 − 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 0.688·19-s − 0.223·20-s + 0.436·21-s − 0.213·22-s − 1.87·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−12.80895195999698, −12.46461346637314, −11.61584118399928, −11.51458068669716, −11.04909154869824, −10.65576085446065, −10.03009969986010, −9.624611964749945, −9.258681010354509, −8.468158169840040, −8.105318779888009, −7.598875250288813, −6.988230819204247, −6.652757237769425, −6.147131604690025, −5.884954797327693, −5.221653132198015, −4.568919603480137, −4.356494092017217, −3.760592350652094, −3.378606837563602, −2.494523440019707, −2.169495362366209, −1.611976247366391, −0.4498366193843025, 0,
0.4498366193843025, 1.611976247366391, 2.169495362366209, 2.494523440019707, 3.378606837563602, 3.760592350652094, 4.356494092017217, 4.568919603480137, 5.221653132198015, 5.884954797327693, 6.147131604690025, 6.652757237769425, 6.988230819204247, 7.598875250288813, 8.105318779888009, 8.468158169840040, 9.258681010354509, 9.624611964749945, 10.03009969986010, 10.65576085446065, 11.04909154869824, 11.51458068669716, 11.61584118399928, 12.46461346637314, 12.80895195999698
