| L(s) = 1 | − 4·2-s − 8·3-s + 16·4-s + 32·6-s + 49·7-s − 64·8-s − 179·9-s − 340·11-s − 128·12-s + 294·13-s − 196·14-s + 256·16-s − 1.22e3·17-s + 716·18-s + 2.43e3·19-s − 392·21-s + 1.36e3·22-s − 2.00e3·23-s + 512·24-s − 1.17e3·26-s + 3.37e3·27-s + 784·28-s − 6.74e3·29-s + 8.85e3·31-s − 1.02e3·32-s + 2.72e3·33-s + 4.90e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.513·3-s + 1/2·4-s + 0.362·6-s + 0.377·7-s − 0.353·8-s − 0.736·9-s − 0.847·11-s − 0.256·12-s + 0.482·13-s − 0.267·14-s + 1/4·16-s − 1.02·17-s + 0.520·18-s + 1.54·19-s − 0.193·21-s + 0.599·22-s − 0.788·23-s + 0.181·24-s − 0.341·26-s + 0.891·27-s + 0.188·28-s − 1.48·29-s + 1.65·31-s − 0.176·32-s + 0.434·33-s + 0.727·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8055793373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8055793373\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 8 T + p^{5} T^{2} \) |
| 11 | \( 1 + 340 T + p^{5} T^{2} \) |
| 13 | \( 1 - 294 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1226 T + p^{5} T^{2} \) |
| 19 | \( 1 - 128 p T + p^{5} T^{2} \) |
| 23 | \( 1 + 2000 T + p^{5} T^{2} \) |
| 29 | \( 1 + 6746 T + p^{5} T^{2} \) |
| 31 | \( 1 - 8856 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14574 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8108 T + p^{5} T^{2} \) |
| 47 | \( 1 - 312 T + p^{5} T^{2} \) |
| 53 | \( 1 - 14634 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27656 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34338 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12316 T + p^{5} T^{2} \) |
| 71 | \( 1 - 520 p T + p^{5} T^{2} \) |
| 73 | \( 1 - 61718 T + p^{5} T^{2} \) |
| 79 | \( 1 + 64752 T + p^{5} T^{2} \) |
| 83 | \( 1 - 77056 T + p^{5} T^{2} \) |
| 89 | \( 1 + 8166 T + p^{5} T^{2} \) |
| 97 | \( 1 + 20650 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.69330623405008690013491896581, −9.815580455424219294223992092910, −8.692193877229756620831147752315, −8.004208742272404224038079059513, −6.91377672584541152682426676656, −5.82780692740114541580765307146, −4.97458393679459304453040076149, −3.30636520828831135217683131197, −2.00199355978845180534823333436, −0.54193408862709128523028195737,
0.54193408862709128523028195737, 2.00199355978845180534823333436, 3.30636520828831135217683131197, 4.97458393679459304453040076149, 5.82780692740114541580765307146, 6.91377672584541152682426676656, 8.004208742272404224038079059513, 8.692193877229756620831147752315, 9.815580455424219294223992092910, 10.69330623405008690013491896581
