| L(s) = 1 | + 5·5-s + 9.46·7-s + 49.0·11-s + 55.4·13-s − 27.9·17-s + 26.3·19-s + 9.53·23-s + 25·25-s + 218.·29-s − 158.·31-s + 47.3·35-s + 189.·37-s − 246.·41-s + 40.2·43-s − 213.·47-s − 253.·49-s − 283.·53-s + 245.·55-s − 92·59-s + 370.·61-s + 277.·65-s + 1.08e3·67-s + 333.·71-s − 606.·73-s + 464.·77-s − 44.3·79-s − 107.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.511·7-s + 1.34·11-s + 1.18·13-s − 0.399·17-s + 0.318·19-s + 0.0864·23-s + 0.200·25-s + 1.39·29-s − 0.920·31-s + 0.228·35-s + 0.842·37-s − 0.937·41-s + 0.142·43-s − 0.664·47-s − 0.738·49-s − 0.734·53-s + 0.601·55-s − 0.203·59-s + 0.776·61-s + 0.528·65-s + 1.98·67-s + 0.557·71-s − 0.972·73-s + 0.687·77-s − 0.0631·79-s − 0.142·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.003853978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.003853978\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 9.46T + 343T^{2} \) |
| 11 | \( 1 - 49.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 27.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 9.53T + 1.21e4T^{2} \) |
| 29 | \( 1 - 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 246.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 40.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 213.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 92T + 2.05e5T^{2} \) |
| 61 | \( 1 - 370.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.08e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 606.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 44.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 107.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 257.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.31e3T + 9.12e5T^{2} \) |
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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
−9.400941184565015510979159469290, −8.734278928703078546158023486228, −7.995159477919477882055771280876, −6.74915740501309006672081450449, −6.27689104271426081196683858938, −5.19677863524265578883424733608, −4.21839909669030073962912585538, −3.27860737584968292133941066063, −1.85855150732230445912457256292, −0.992124289239819836623910230322,
0.992124289239819836623910230322, 1.85855150732230445912457256292, 3.27860737584968292133941066063, 4.21839909669030073962912585538, 5.19677863524265578883424733608, 6.27689104271426081196683858938, 6.74915740501309006672081450449, 7.995159477919477882055771280876, 8.734278928703078546158023486228, 9.400941184565015510979159469290
