| L(s) = 1 | + 3-s + 9-s + 11-s − 4·17-s − 8·19-s − 5·25-s + 27-s − 4·29-s + 6·31-s + 33-s − 6·37-s − 6·41-s + 2·43-s + 8·47-s − 7·49-s − 4·51-s − 6·53-s − 8·57-s + 14·61-s + 14·67-s + 4·71-s − 6·73-s − 5·75-s − 10·79-s + 81-s − 12·83-s − 4·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.970·17-s − 1.83·19-s − 25-s + 0.192·27-s − 0.742·29-s + 1.07·31-s + 0.174·33-s − 0.986·37-s − 0.937·41-s + 0.304·43-s + 1.16·47-s − 49-s − 0.560·51-s − 0.824·53-s − 1.05·57-s + 1.79·61-s + 1.71·67-s + 0.474·71-s − 0.702·73-s − 0.577·75-s − 1.12·79-s + 1/9·81-s − 1.31·83-s − 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.75120818304188, −12.57987357170329, −11.70104921233088, −11.42087567729672, −11.04341207691706, −10.34735646111833, −10.05307568169363, −9.632846838365516, −8.979085674133286, −8.521567617914091, −8.446153949067623, −7.790032985751517, −7.131608634119194, −6.837349987450916, −6.297876948409512, −5.886270994142694, −5.224573843794646, −4.562752798520020, −4.244967415053531, −3.747478143989912, −3.216523410511790, −2.490356007220721, −1.997149711005976, −1.713204951612716, −0.6922024418480748, 0,
0.6922024418480748, 1.713204951612716, 1.997149711005976, 2.490356007220721, 3.216523410511790, 3.747478143989912, 4.244967415053531, 4.562752798520020, 5.224573843794646, 5.886270994142694, 6.297876948409512, 6.837349987450916, 7.131608634119194, 7.790032985751517, 8.446153949067623, 8.521567617914091, 8.979085674133286, 9.632846838365516, 10.05307568169363, 10.34735646111833, 11.04341207691706, 11.42087567729672, 11.70104921233088, 12.57987357170329, 12.75120818304188
