L(s) = 1 | − 1.08·3-s − 2.09i·5-s + 3.72·7-s − 7.81·9-s + 6.64i·11-s + 9.32·13-s + 2.27i·15-s − 2.02·17-s + (−17.3 − 7.78i)19-s − 4.04·21-s + 7.73·23-s + 20.6·25-s + 18.2·27-s − 35.3·29-s + 37.0i·31-s + ⋯ |
L(s) = 1 | − 0.362·3-s − 0.418i·5-s + 0.531·7-s − 0.868·9-s + 0.603i·11-s + 0.717·13-s + 0.151i·15-s − 0.118·17-s + (−0.912 − 0.409i)19-s − 0.192·21-s + 0.336·23-s + 0.824·25-s + 0.677·27-s − 1.21·29-s + 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.133049092\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.133049092\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (17.3 + 7.78i)T \) |
good | 3 | \( 1 + 1.08T + 9T^{2} \) |
| 5 | \( 1 + 2.09iT - 25T^{2} \) |
| 7 | \( 1 - 3.72T + 49T^{2} \) |
| 11 | \( 1 - 6.64iT - 121T^{2} \) |
| 13 | \( 1 - 9.32T + 169T^{2} \) |
| 17 | \( 1 + 2.02T + 289T^{2} \) |
| 23 | \( 1 - 7.73T + 529T^{2} \) |
| 29 | \( 1 + 35.3T + 841T^{2} \) |
| 31 | \( 1 - 37.0iT - 961T^{2} \) |
| 37 | \( 1 + 33.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 28.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 73.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 59.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 60.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 18.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 91.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 83.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 5.17iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 39.3iT - 9.40e3T^{2} \) |
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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
−9.652647920626848414047731377559, −8.699049617471602789234179136088, −8.379696186221995917258030550244, −7.16772639429479423478356192343, −6.38892170369395158034773824297, −5.36196400845245477635825965472, −4.78114219263328370358079497937, −3.65515896953526595959508667576, −2.37336976090120444040156596227, −1.12451194951797321778036305577,
0.38739727193444337900446195206, 1.92836181210892388994651348682, 3.12479114490457989799350908497, 4.08603156795812566825262846654, 5.29033053757902428902593376367, 5.96014185325783785561412806146, 6.73598836633057944191399385804, 7.81194976799383543499204125970, 8.563147141119231304493545915662, 9.188137313897244266404556429065