L(s) = 1 | + (0.202 − 1.97i)2-s + (−0.930 − 0.366i)3-s + (−2.86 − 0.595i)4-s + (−0.911 + 1.75i)6-s + (−1.15 + 3.63i)8-s + (0.730 + 0.682i)9-s + (2.44 + 1.60i)12-s + (4.25 + 1.84i)16-s + (−0.113 − 0.660i)17-s + (1.49 − 1.30i)18-s + (1.93 − 0.266i)19-s + (1.25 − 0.129i)23-s + (2.40 − 2.95i)24-s + (−0.429 − 0.903i)27-s + (−0.650 + 1.49i)31-s + (2.63 − 4.67i)32-s + ⋯ |
L(s) = 1 | + (0.202 − 1.97i)2-s + (−0.930 − 0.366i)3-s + (−2.86 − 0.595i)4-s + (−0.911 + 1.75i)6-s + (−1.15 + 3.63i)8-s + (0.730 + 0.682i)9-s + (2.44 + 1.60i)12-s + (4.25 + 1.84i)16-s + (−0.113 − 0.660i)17-s + (1.49 − 1.30i)18-s + (1.93 − 0.266i)19-s + (1.25 − 0.129i)23-s + (2.40 − 2.95i)24-s + (−0.429 − 0.903i)27-s + (−0.650 + 1.49i)31-s + (2.63 − 4.67i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8375186174\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8375186174\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.930 + 0.366i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (0.971 - 0.236i)T \) |
good | 2 | \( 1 + (-0.202 + 1.97i)T + (-0.979 - 0.203i)T^{2} \) |
| 7 | \( 1 + (-0.631 - 0.775i)T^{2} \) |
| 11 | \( 1 + (-0.334 + 0.942i)T^{2} \) |
| 13 | \( 1 + (-0.136 - 0.990i)T^{2} \) |
| 17 | \( 1 + (0.113 + 0.660i)T + (-0.942 + 0.334i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 0.266i)T + (0.962 - 0.269i)T^{2} \) |
| 23 | \( 1 + (-1.25 + 0.129i)T + (0.979 - 0.203i)T^{2} \) |
| 29 | \( 1 + (0.990 + 0.136i)T^{2} \) |
| 31 | \( 1 + (0.650 - 1.49i)T + (-0.682 - 0.730i)T^{2} \) |
| 37 | \( 1 + (0.887 + 0.460i)T^{2} \) |
| 41 | \( 1 + (-0.576 - 0.816i)T^{2} \) |
| 43 | \( 1 + (0.398 - 0.917i)T^{2} \) |
| 53 | \( 1 + (-1.30 + 0.412i)T + (0.816 - 0.576i)T^{2} \) |
| 59 | \( 1 + (-0.917 + 0.398i)T^{2} \) |
| 61 | \( 1 + (-0.116 - 0.0709i)T + (0.460 + 0.887i)T^{2} \) |
| 67 | \( 1 + (-0.631 + 0.775i)T^{2} \) |
| 71 | \( 1 + (-0.203 - 0.979i)T^{2} \) |
| 73 | \( 1 + (-0.997 - 0.0682i)T^{2} \) |
| 79 | \( 1 + (0.311 + 1.11i)T + (-0.854 + 0.519i)T^{2} \) |
| 83 | \( 1 + (-0.156 + 0.906i)T + (-0.942 - 0.334i)T^{2} \) |
| 89 | \( 1 + (0.962 + 0.269i)T^{2} \) |
| 97 | \( 1 + (0.730 + 0.682i)T^{2} \) |
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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
−8.842851469044339551240303892149, −7.72808062446952253316142053123, −6.92910480181636774626699905462, −5.64687430565459269113726972140, −5.12740143354736529034858047901, −4.57846295492353224426761345355, −3.43396975895255122359845599298, −2.76907961259388126470488777140, −1.59396389375125726128275501227, −0.803611486932574751570089729054,
0.926411135918819361806622702022, 3.40164370008083773927929627452, 4.06555976832186340605114859470, 4.96126974280160786243975151911, 5.51133709458929931935819644770, 5.99478199291098093135392484057, 6.93740366403288899681260876270, 7.30387451168100807339282392834, 8.121394538021442035367754441448, 8.988457298031974891593401596531