L(s) = 1 | − 1.65·2-s + 3-s + 0.723·4-s + 4.43·5-s − 1.65·6-s − 3.18·7-s + 2.10·8-s + 9-s − 7.31·10-s + 11-s + 0.723·12-s + 5.26·14-s + 4.43·15-s − 4.92·16-s − 1.25·17-s − 1.65·18-s + 5.97·19-s + 3.20·20-s − 3.18·21-s − 1.65·22-s + 6.94·23-s + 2.10·24-s + 14.6·25-s + 27-s − 2.30·28-s + 5.43·29-s − 7.31·30-s + ⋯ |
L(s) = 1 | − 1.16·2-s + 0.577·3-s + 0.361·4-s + 1.98·5-s − 0.673·6-s − 1.20·7-s + 0.744·8-s + 0.333·9-s − 2.31·10-s + 0.301·11-s + 0.208·12-s + 1.40·14-s + 1.14·15-s − 1.23·16-s − 0.303·17-s − 0.389·18-s + 1.37·19-s + 0.717·20-s − 0.695·21-s − 0.351·22-s + 1.44·23-s + 0.429·24-s + 2.92·25-s + 0.192·27-s − 0.435·28-s + 1.00·29-s − 1.33·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.903312805\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.903312805\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 5 | \( 1 - 4.43T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 17 | \( 1 + 1.25T + 17T^{2} \) |
| 19 | \( 1 - 5.97T + 19T^{2} \) |
| 23 | \( 1 - 6.94T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 - 2.52T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 0.983T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 + 2.73T + 59T^{2} \) |
| 61 | \( 1 + 2.73T + 61T^{2} \) |
| 67 | \( 1 + 9.44T + 67T^{2} \) |
| 71 | \( 1 - 6.30T + 71T^{2} \) |
| 73 | \( 1 - 1.95T + 73T^{2} \) |
| 79 | \( 1 + 9.68T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 + 2.64T + 89T^{2} \) |
| 97 | \( 1 - 2.66T + 97T^{2} \) |
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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
−8.592201509860797587378696311245, −7.46257466304353667362349534299, −6.64591245676032661362053108769, −6.43588858043326419754875003981, −5.29636536562714305347493794326, −4.66429984621287100090779533358, −3.18467257190384813324727230509, −2.71669046473187743720664046702, −1.62809945324492102102345549169, −0.931082693821154427987442061988,
0.931082693821154427987442061988, 1.62809945324492102102345549169, 2.71669046473187743720664046702, 3.18467257190384813324727230509, 4.66429984621287100090779533358, 5.29636536562714305347493794326, 6.43588858043326419754875003981, 6.64591245676032661362053108769, 7.46257466304353667362349534299, 8.592201509860797587378696311245