L(s) = 1 | + (−1.21 + 2.09i)5-s + (−2.56 + 0.658i)7-s + (2.35 + 4.07i)11-s + (1.71 + 2.96i)13-s + (0.851 − 1.47i)17-s + (−0.641 − 1.11i)19-s + (−0.562 + 0.974i)23-s + (−0.430 − 0.746i)25-s + (−2.35 + 4.07i)29-s − 3.42·31-s + (1.72 − 6.17i)35-s + (−4.27 − 7.40i)37-s + (1.85 + 3.21i)41-s + (−2.77 + 4.80i)43-s + 11.8·47-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.937i)5-s + (−0.968 + 0.249i)7-s + (0.709 + 1.22i)11-s + (0.474 + 0.821i)13-s + (0.206 − 0.357i)17-s + (−0.147 − 0.254i)19-s + (−0.117 + 0.203i)23-s + (−0.0861 − 0.149i)25-s + (−0.436 + 0.756i)29-s − 0.614·31-s + (0.290 − 1.04i)35-s + (−0.702 − 1.21i)37-s + (0.290 + 0.502i)41-s + (−0.422 + 0.732i)43-s + 1.72·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7290078871\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7290078871\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.56 - 0.658i)T \) |
good | 5 | \( 1 + (1.21 - 2.09i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.35 - 4.07i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.71 - 2.96i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.851 + 1.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.641 + 1.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.562 - 0.974i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.35 - 4.07i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 + (4.27 + 7.40i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.85 - 3.21i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.77 - 4.80i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + (-5.13 + 8.88i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 4.12T + 59T^{2} \) |
| 61 | \( 1 + 9.24T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + (1.06 - 1.85i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + (-4.21 + 7.29i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.04 - 13.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.12 - 14.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.212349590987730715686763480699, −8.989866887587230540141348728181, −7.51959014414942919356857355725, −7.10131549824887963654916750425, −6.51694999358424265956520437937, −5.61318963431816277589101068084, −4.37478347848738606756499919504, −3.68767062410425651453073321205, −2.83842132311884257877032828803, −1.70141834636067025403141872588,
0.26950121376075235611103546419, 1.26076312238113980114004971604, 2.97857613145833086994926594131, 3.73091622869532737835609344346, 4.41018225758973032244684998825, 5.76836904871349476009330967404, 6.00940236084481602884015887771, 7.16278862598787907142472288147, 7.973225082313703173868577023513, 8.817413542073001686054376134139