| L(s) = 1 | + 3-s + 5-s − 2·7-s − 2·9-s − 11-s − 5·13-s + 15-s + 17-s + 5·19-s − 2·21-s + 6·23-s + 25-s − 5·27-s − 29-s + 5·31-s − 33-s − 2·35-s + 8·37-s − 5·39-s − 2·41-s − 6·43-s − 2·45-s − 5·47-s − 3·49-s + 51-s + 5·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s − 1.38·13-s + 0.258·15-s + 0.242·17-s + 1.14·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.962·27-s − 0.185·29-s + 0.898·31-s − 0.174·33-s − 0.338·35-s + 1.31·37-s − 0.800·39-s − 0.312·41-s − 0.914·43-s − 0.298·45-s − 0.729·47-s − 3/7·49-s + 0.140·51-s + 0.686·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.61170305051232, −14.03489191984716, −13.59737434919602, −12.98447788615071, −12.85170646355337, −11.88652311546685, −11.69016805046124, −11.07647654303421, −10.15729913533595, −9.942017542607850, −9.537644059884141, −8.889311076465843, −8.521661335219163, −7.673162164856030, −7.395695481069499, −6.774882962652428, −6.073591194015997, −5.591018343803542, −4.957147301174199, −4.503459086520279, −3.419088531230288, −2.988192541336009, −2.684144250317250, −1.870645829677187, −0.9088287404746563, 0,
0.9088287404746563, 1.870645829677187, 2.684144250317250, 2.988192541336009, 3.419088531230288, 4.503459086520279, 4.957147301174199, 5.591018343803542, 6.073591194015997, 6.774882962652428, 7.395695481069499, 7.673162164856030, 8.521661335219163, 8.889311076465843, 9.537644059884141, 9.942017542607850, 10.15729913533595, 11.07647654303421, 11.69016805046124, 11.88652311546685, 12.85170646355337, 12.98447788615071, 13.59737434919602, 14.03489191984716, 14.61170305051232
