| L(s) = 1 | − 207.·3-s − 631. i·5-s + (−5.15e3 + 3.71e3i)7-s + 2.34e4·9-s − 8.48e4i·11-s + 7.82e4i·13-s + 1.31e5i·15-s + 6.16e5i·17-s + 4.79e5·19-s + (1.07e6 − 7.71e5i)21-s − 5.99e5i·23-s + 1.55e6·25-s − 7.79e5·27-s + 3.67e6·29-s − 4.75e6·31-s + ⋯ |
| L(s) = 1 | − 1.48·3-s − 0.451i·5-s + (−0.811 + 0.584i)7-s + 1.19·9-s − 1.74i·11-s + 0.759i·13-s + 0.668i·15-s + 1.79i·17-s + 0.844·19-s + (1.20 − 0.865i)21-s − 0.446i·23-s + 0.795·25-s − 0.282·27-s + 0.963·29-s − 0.924·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2888928152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2888928152\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (5.15e3 - 3.71e3i)T \) |
| good | 3 | \( 1 + 207.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 631. iT - 1.95e6T^{2} \) |
| 11 | \( 1 + 8.48e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 7.82e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 6.16e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 4.79e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.99e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 - 3.67e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.75e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.17e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.88e5iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 3.80e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 1.49e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.09e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.94e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.24e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 1.52e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 7.56e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + 2.09e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 4.17e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 6.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.52e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + 7.61e8iT - 7.60e17T^{2} \) |
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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
−11.39151170084563049068440442418, −10.68433377794302869100083029831, −9.347771344826923852310181272465, −8.267304063804192289885945533981, −6.33637091227227531683085790408, −6.05766436311000471539606555354, −4.80742778488007928821468018983, −3.27812920840405260580477398826, −1.24279511418198955158742994636, −0.12370040426798215454213897694,
0.964142382906974562255019138237, 2.90007124214799828101943151809, 4.56262153202681937190801087316, 5.51795169387147499479845783085, 6.90381853490286219111551693463, 7.28838990843881272822689892968, 9.588165793561334048751255221643, 10.20811298351080129930438647751, 11.23858417441461071943713548780, 12.18742508894109645570013222511
