| L(s) = 1 | + i·2-s − 4-s + (−2 − i)5-s + 4i·7-s − i·8-s + (1 − 2i)10-s + 6·11-s + i·13-s − 4·14-s + 16-s − 4i·17-s − 2·19-s + (2 + i)20-s + 6i·22-s + 6i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 1.51i·7-s − 0.353i·8-s + (0.316 − 0.632i)10-s + 1.80·11-s + 0.277i·13-s − 1.06·14-s + 0.250·16-s − 0.970i·17-s − 0.458·19-s + (0.447 + 0.223i)20-s + 1.27i·22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.007861202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007861202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−9.567518373047799809538684250061, −9.171658757631498117001698371959, −8.591708887486207961250456715206, −7.66816962489719701244400729539, −6.80580627847705756631003581304, −5.96740314632362868663684570658, −5.09112032500399629308610618367, −4.18387556564618950359264905326, −3.20411803066284084213875052350, −1.52253880076688652049512571959,
0.46842484086153357302250841885, 1.78897781410393362210516794694, 3.56866444562316507623612059497, 3.82990033815388728334514996054, 4.65753041938662384480582510665, 6.31767093213638303793059446864, 6.93002502958946787580321173830, 7.84799264070709687793616234040, 8.641452540613260864655540861877, 9.570953681673911956927210763053
