| L(s) = 1 | + 23.1i·5-s + (−6.45 − 48.5i)7-s + 191.·11-s − 48.5i·13-s + 181. i·17-s − 599. i·19-s − 469.·23-s + 86.8·25-s + 338.·29-s − 267. i·31-s + (1.12e3 − 149. i)35-s − 668.·37-s − 1.32e3i·41-s − 1.94e3·43-s + 2.93e3i·47-s + ⋯ |
| L(s) = 1 | + 0.927i·5-s + (−0.131 − 0.991i)7-s + 1.58·11-s − 0.287i·13-s + 0.629i·17-s − 1.66i·19-s − 0.887·23-s + 0.138·25-s + 0.402·29-s − 0.278i·31-s + (0.919 − 0.122i)35-s − 0.488·37-s − 0.787i·41-s − 1.04·43-s + 1.32i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.767702682\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.767702682\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (6.45 + 48.5i)T \) |
| good | 5 | \( 1 - 23.1iT - 625T^{2} \) |
| 11 | \( 1 - 191.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 48.5iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 181. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 599. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 469.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 338.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 267. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 668.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.32e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.94e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.93e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.46e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.73e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 246. iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.07e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.27e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 7.10e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 7.01e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.44e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 2.13e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 5.89e3iT - 8.85e7T^{2} \) |
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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
−9.299521043292785213846484943180, −8.394782692109204828056502114979, −7.26918713699157705138534316545, −6.76970425332806891056626612093, −6.07871217292610781745432857027, −4.62413563135158613937228161203, −3.82042950646673874149593499115, −2.95711517150833715604480851808, −1.59748663300775801871278148219, −0.40340189486343437782803579759,
1.12565163615155350134488804994, 1.97605869537903198101483892750, 3.41943785820701800454937557425, 4.33245871120119556641617292723, 5.32642409762674494789294475960, 6.11990400602217175542699092068, 6.95134463195477218687750812091, 8.280243443676521339568610326761, 8.702366291606177787521723022198, 9.508496781836379317209998476078
