L(s) = 1 | + 2·5-s + 2·13-s + 2·17-s − 4·19-s − 8·23-s − 25-s + 6·29-s − 8·31-s + 6·37-s − 6·41-s + 4·43-s − 7·49-s + 2·53-s + 4·59-s + 2·61-s + 4·65-s + 4·67-s + 8·71-s − 10·73-s − 8·79-s + 4·83-s + 4·85-s + 6·89-s − 8·95-s + 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s − 49-s + 0.274·53-s + 0.520·59-s + 0.256·61-s + 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 0.439·83-s + 0.433·85-s + 0.635·89-s − 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.24688640800094, −15.65379297728913, −14.90336869465901, −14.38593959725412, −13.99230880412379, −13.32935866868449, −12.94968962861202, −12.25511467641868, −11.74497794529060, −11.00249003564484, −10.48590982347771, −9.872719359848364, −9.534217458881954, −8.694901520701798, −8.222404795939091, −7.612582757251573, −6.724723159923009, −6.202786469001169, −5.735595441825211, −5.076566067543094, −4.162527404982158, −3.672494877790323, −2.648356895289312, −2.000469175524955, −1.279461821856717, 0,
1.279461821856717, 2.000469175524955, 2.648356895289312, 3.672494877790323, 4.162527404982158, 5.076566067543094, 5.735595441825211, 6.202786469001169, 6.724723159923009, 7.612582757251573, 8.222404795939091, 8.694901520701798, 9.534217458881954, 9.872719359848364, 10.48590982347771, 11.00249003564484, 11.74497794529060, 12.25511467641868, 12.94968962861202, 13.32935866868449, 13.99230880412379, 14.38593959725412, 14.90336869465901, 15.65379297728913, 16.24688640800094