| L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s − 7-s − 3·8-s + 9-s − 10-s + 12-s + 4·13-s − 14-s + 15-s − 16-s + 18-s + 2·19-s + 20-s + 21-s + 2·23-s + 3·24-s + 25-s + 4·26-s − 27-s + 28-s − 8·29-s + 30-s − 4·31-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 − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s + 1.10·13-s − 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.218·21-s + 0.417·23-s + 0.612·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.188·28-s − 1.48·29-s + 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.37098358094604, −14.81964295761980, −14.30179194489080, −13.70134693296639, −13.20934822109678, −12.78453216046732, −12.33966966563361, −11.75915510608613, −11.07766418166346, −10.89246180168007, −10.00249725188109, −9.359863466384801, −9.004537210032391, −8.336949191912584, −7.640474228982026, −7.018139900946631, −6.359893881710696, −5.763674395490842, −5.360783602908439, −4.690169926035935, −3.946802247338131, −3.588135774204235, −2.975147092171356, −1.825165231156250, −0.8585117660668539, 0,
0.8585117660668539, 1.825165231156250, 2.975147092171356, 3.588135774204235, 3.946802247338131, 4.690169926035935, 5.360783602908439, 5.763674395490842, 6.359893881710696, 7.018139900946631, 7.640474228982026, 8.336949191912584, 9.004537210032391, 9.359863466384801, 10.00249725188109, 10.89246180168007, 11.07766418166346, 11.75915510608613, 12.33966966563361, 12.78453216046732, 13.20934822109678, 13.70134693296639, 14.30179194489080, 14.81964295761980, 15.37098358094604
