| L(s) = 1 | − 8·3-s − 208·7-s − 179·9-s + 536·11-s − 694·13-s + 1.27e3·17-s − 1.11e3·19-s + 1.66e3·21-s + 3.21e3·23-s + 3.37e3·27-s + 2.91e3·29-s + 2.62e3·31-s − 4.28e3·33-s + 9.45e3·37-s + 5.55e3·39-s + 170·41-s − 1.99e4·43-s + 32·47-s + 2.64e4·49-s − 1.02e4·51-s + 2.21e4·53-s + 8.89e3·57-s − 4.14e4·59-s + 1.54e4·61-s + 3.72e4·63-s − 2.07e4·67-s − 2.57e4·69-s + ⋯ |
| L(s) = 1 | − 0.513·3-s − 1.60·7-s − 0.736·9-s + 1.33·11-s − 1.13·13-s + 1.07·17-s − 0.706·19-s + 0.823·21-s + 1.26·23-s + 0.891·27-s + 0.644·29-s + 0.490·31-s − 0.685·33-s + 1.13·37-s + 0.584·39-s + 0.0157·41-s − 1.64·43-s + 0.00211·47-s + 1.57·49-s − 0.550·51-s + 1.08·53-s + 0.362·57-s − 1.55·59-s + 0.532·61-s + 1.18·63-s − 0.564·67-s − 0.650·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{5} T^{2} \) |
| 7 | \( 1 + 208 T + p^{5} T^{2} \) |
| 11 | \( 1 - 536 T + p^{5} T^{2} \) |
| 13 | \( 1 + 694 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1278 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1112 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3216 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2918 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2624 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9458 T + p^{5} T^{2} \) |
| 41 | \( 1 - 170 T + p^{5} T^{2} \) |
| 43 | \( 1 + 19928 T + p^{5} T^{2} \) |
| 47 | \( 1 - 32 T + p^{5} T^{2} \) |
| 53 | \( 1 - 22178 T + p^{5} T^{2} \) |
| 59 | \( 1 + 41480 T + p^{5} T^{2} \) |
| 61 | \( 1 - 15462 T + p^{5} T^{2} \) |
| 67 | \( 1 + 20744 T + p^{5} T^{2} \) |
| 71 | \( 1 + 28592 T + p^{5} T^{2} \) |
| 73 | \( 1 - 53670 T + p^{5} T^{2} \) |
| 79 | \( 1 - 69152 T + p^{5} T^{2} \) |
| 83 | \( 1 + 37800 T + p^{5} T^{2} \) |
| 89 | \( 1 + 126806 T + p^{5} T^{2} \) |
| 97 | \( 1 + 62290 T + p^{5} T^{2} \) |
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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
−9.310478615354958693585063333332, −8.344503318597969240171330773407, −7.02714030013747150249677724349, −6.49813429722211491263625048226, −5.71880858057772144170596067315, −4.63141331100543662195376649023, −3.42053391757509699858042639949, −2.68457245892968146571124372700, −0.990589169647647563343547327074, 0,
0.990589169647647563343547327074, 2.68457245892968146571124372700, 3.42053391757509699858042639949, 4.63141331100543662195376649023, 5.71880858057772144170596067315, 6.49813429722211491263625048226, 7.02714030013747150249677724349, 8.344503318597969240171330773407, 9.310478615354958693585063333332
