| L(s) = 1 | − 4·2-s − 10·3-s + 16·4-s − 84·5-s + 40·6-s − 64·8-s − 143·9-s + 336·10-s − 336·11-s − 160·12-s − 584·13-s + 840·15-s + 256·16-s + 1.45e3·17-s + 572·18-s − 470·19-s − 1.34e3·20-s + 1.34e3·22-s − 4.20e3·23-s + 640·24-s + 3.93e3·25-s + 2.33e3·26-s + 3.86e3·27-s + 4.86e3·29-s − 3.36e3·30-s + 7.37e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.641·3-s + 1/2·4-s − 1.50·5-s + 0.453·6-s − 0.353·8-s − 0.588·9-s + 1.06·10-s − 0.837·11-s − 0.320·12-s − 0.958·13-s + 0.963·15-s + 1/4·16-s + 1.22·17-s + 0.416·18-s − 0.298·19-s − 0.751·20-s + 0.592·22-s − 1.65·23-s + 0.226·24-s + 1.25·25-s + 0.677·26-s + 1.01·27-s + 1.07·29-s − 0.681·30-s + 1.37·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3616055471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3616055471\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 10 T + p^{5} T^{2} \) |
| 5 | \( 1 + 84 T + p^{5} T^{2} \) |
| 11 | \( 1 + 336 T + p^{5} T^{2} \) |
| 13 | \( 1 + 584 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1458 T + p^{5} T^{2} \) |
| 19 | \( 1 + 470 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4866 T + p^{5} T^{2} \) |
| 31 | \( 1 - 7372 T + p^{5} T^{2} \) |
| 37 | \( 1 - 14330 T + p^{5} T^{2} \) |
| 41 | \( 1 + 6222 T + p^{5} T^{2} \) |
| 43 | \( 1 - 3704 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1812 T + p^{5} T^{2} \) |
| 53 | \( 1 + 37242 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34302 T + p^{5} T^{2} \) |
| 61 | \( 1 + 24476 T + p^{5} T^{2} \) |
| 67 | \( 1 + 17452 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28224 T + p^{5} T^{2} \) |
| 73 | \( 1 + 3602 T + p^{5} T^{2} \) |
| 79 | \( 1 - 42872 T + p^{5} T^{2} \) |
| 83 | \( 1 - 35202 T + p^{5} T^{2} \) |
| 89 | \( 1 + 26730 T + p^{5} T^{2} \) |
| 97 | \( 1 - 16978 T + p^{5} T^{2} \) |
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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
−12.28333482920652058461842308263, −11.94328115537191404128648513591, −10.86863502387414363180937622804, −9.877994411032341752796002500442, −8.148453351104689951083911860489, −7.74913221738114551207573207832, −6.18239302511479670313313305972, −4.67453725403674575218394030923, −2.92288690615189720289646137799, −0.47335253516650937983775240582,
0.47335253516650937983775240582, 2.92288690615189720289646137799, 4.67453725403674575218394030923, 6.18239302511479670313313305972, 7.74913221738114551207573207832, 8.148453351104689951083911860489, 9.877994411032341752796002500442, 10.86863502387414363180937622804, 11.94328115537191404128648513591, 12.28333482920652058461842308263
