L(s) = 1 | + 2.41·2-s − 2.41·3-s + 3.82·4-s − 5.82·6-s + 2.82·7-s + 4.41·8-s + 2.82·9-s − 0.414·11-s − 9.24·12-s + 3.82·13-s + 6.82·14-s + 2.99·16-s − 0.828·17-s + 6.82·18-s + 6·19-s − 6.82·21-s − 0.999·22-s − 3.65·23-s − 10.6·24-s + 9.24·26-s + 0.414·27-s + 10.8·28-s + 29-s + 10.0·31-s − 1.58·32-s + 0.999·33-s − 1.99·34-s + ⋯ |
L(s) = 1 | + 1.70·2-s − 1.39·3-s + 1.91·4-s − 2.37·6-s + 1.06·7-s + 1.56·8-s + 0.942·9-s − 0.124·11-s − 2.66·12-s + 1.06·13-s + 1.82·14-s + 0.749·16-s − 0.200·17-s + 1.60·18-s + 1.37·19-s − 1.49·21-s − 0.213·22-s − 0.762·23-s − 2.17·24-s + 1.81·26-s + 0.0797·27-s + 2.04·28-s + 0.185·29-s + 1.80·31-s − 0.280·32-s + 0.174·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.962861586\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.962861586\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 - 3.24T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 + 7.65T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
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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
−10.96768570117654535644916363631, −10.04339143715727142108394245380, −8.435793781755063820145050422541, −7.37150055414503612739584494594, −6.29004227832335436230504008703, −5.82510621339743476881731562127, −4.91298639451727018619727516006, −4.39580848726231749862998297714, −3.07712937017362101851030593142, −1.42569662630534432007409229873,
1.42569662630534432007409229873, 3.07712937017362101851030593142, 4.39580848726231749862998297714, 4.91298639451727018619727516006, 5.82510621339743476881731562127, 6.29004227832335436230504008703, 7.37150055414503612739584494594, 8.435793781755063820145050422541, 10.04339143715727142108394245380, 10.96768570117654535644916363631