L(s) = 1 | + 2.53·2-s + 0.652·3-s + 4.41·4-s − 1.34·5-s + 1.65·6-s + 1.53·7-s + 6.10·8-s − 2.57·9-s − 3.41·10-s − 1.18·11-s + 2.87·12-s − 2.71·13-s + 3.87·14-s − 0.879·15-s + 6.63·16-s + 3.87·17-s − 6.51·18-s − 5.94·20-s + 0.999·21-s − 3.00·22-s − 5.06·23-s + 3.98·24-s − 3.18·25-s − 6.87·26-s − 3.63·27-s + 6.75·28-s + 4.65·29-s + ⋯ |
L(s) = 1 | + 1.79·2-s + 0.376·3-s + 2.20·4-s − 0.602·5-s + 0.674·6-s + 0.579·7-s + 2.15·8-s − 0.857·9-s − 1.07·10-s − 0.357·11-s + 0.831·12-s − 0.753·13-s + 1.03·14-s − 0.227·15-s + 1.65·16-s + 0.940·17-s − 1.53·18-s − 1.32·20-s + 0.218·21-s − 0.639·22-s − 1.05·23-s + 0.813·24-s − 0.636·25-s − 1.34·26-s − 0.700·27-s + 1.27·28-s + 0.863·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.567863590\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.567863590\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 \) |
good | 2 | \( 1 - 2.53T + 2T^{2} \) |
| 3 | \( 1 - 0.652T + 3T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 1.18T + 11T^{2} \) |
| 13 | \( 1 + 2.71T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 - 4.65T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 4.10T + 37T^{2} \) |
| 41 | \( 1 - 9.98T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 - 0.573T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 - 3.93T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 + 9.80T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 2.42T + 89T^{2} \) |
| 97 | \( 1 + 7.36T + 97T^{2} \) |
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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
−11.81328802381635186866174799427, −11.02488772387858565329719848489, −9.826649694326291032326070763208, −8.165051593230146577947045230875, −7.63959083352209911361478424563, −6.28526575603015736189868847262, −5.34515168780847009755595846648, −4.43041809116077034329245292354, −3.35345298770707616896232966929, −2.33778935727498860662206296797,
2.33778935727498860662206296797, 3.35345298770707616896232966929, 4.43041809116077034329245292354, 5.34515168780847009755595846648, 6.28526575603015736189868847262, 7.63959083352209911361478424563, 8.165051593230146577947045230875, 9.826649694326291032326070763208, 11.02488772387858565329719848489, 11.81328802381635186866174799427