| L(s) = 1 | + 3-s + 2·5-s + 7-s − 2·9-s − 2·11-s + 13-s + 2·15-s − 6·19-s + 21-s − 23-s − 25-s − 5·27-s − 29-s − 31-s − 2·33-s + 2·35-s + 6·37-s + 39-s + 3·41-s − 4·45-s + 3·47-s + 49-s − 6·53-s − 4·55-s − 6·57-s + 8·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s + 0.516·15-s − 1.37·19-s + 0.218·21-s − 0.208·23-s − 1/5·25-s − 0.962·27-s − 0.185·29-s − 0.179·31-s − 0.348·33-s + 0.338·35-s + 0.986·37-s + 0.160·39-s + 0.468·41-s − 0.596·45-s + 0.437·47-s + 1/7·49-s − 0.824·53-s − 0.539·55-s − 0.794·57-s + 1.04·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.95031281518099, −16.38836345127907, −15.62989851644835, −15.06847388342920, −14.40743947923022, −14.21138818800045, −13.35145948332259, −13.09757705783944, −12.43590997857899, −11.44767838883151, −11.15268148625401, −10.32352457846236, −9.877402445936206, −9.112038042402011, −8.594154865826952, −8.086246525088619, −7.418803721684858, −6.514423905777972, −5.837492507342590, −5.456708058584579, −4.461249718877446, −3.795292666071134, −2.701838170924305, −2.330923084869619, −1.449661190746120, 0,
1.449661190746120, 2.330923084869619, 2.701838170924305, 3.795292666071134, 4.461249718877446, 5.456708058584579, 5.837492507342590, 6.514423905777972, 7.418803721684858, 8.086246525088619, 8.594154865826952, 9.112038042402011, 9.877402445936206, 10.32352457846236, 11.15268148625401, 11.44767838883151, 12.43590997857899, 13.09757705783944, 13.35145948332259, 14.21138818800045, 14.40743947923022, 15.06847388342920, 15.62989851644835, 16.38836345127907, 16.95031281518099
