| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s − 2·9-s + 10-s − 12-s − 3·13-s + 14-s + 15-s + 16-s − 7·17-s + 2·18-s − 19-s − 20-s + 21-s − 5·23-s + 24-s + 25-s + 3·26-s + 5·27-s − 28-s − 5·29-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 − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.288·12-s − 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.69·17-s + 0.471·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s − 1.04·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.962·27-s − 0.188·28-s − 0.928·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−11.80119378341820585349833476455, −11.14134483144774932352515769621, −10.13250645401623025957365197954, −9.052009488114563992693077068684, −8.079561289589277419191527740912, −6.88551821385603277867593120567, −5.93621501809148728100755282492, −4.42087197415110762692909845352, −2.58032194053261503171562760419, 0,
2.58032194053261503171562760419, 4.42087197415110762692909845352, 5.93621501809148728100755282492, 6.88551821385603277867593120567, 8.079561289589277419191527740912, 9.052009488114563992693077068684, 10.13250645401623025957365197954, 11.14134483144774932352515769621, 11.80119378341820585349833476455
