| L(s) = 1 | + 3·7-s + 11-s − 13-s − 3·17-s + 2·19-s − 5·23-s + 6·29-s − 10·31-s − 5·37-s − 3·41-s + 4·43-s − 6·47-s + 2·49-s + 5·53-s − 8·59-s + 61-s + 12·67-s + 71-s + 10·73-s + 3·77-s + 79-s − 89-s − 3·91-s − 3·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.301·11-s − 0.277·13-s − 0.727·17-s + 0.458·19-s − 1.04·23-s + 1.11·29-s − 1.79·31-s − 0.821·37-s − 0.468·41-s + 0.609·43-s − 0.875·47-s + 2/7·49-s + 0.686·53-s − 1.04·59-s + 0.128·61-s + 1.46·67-s + 0.118·71-s + 1.17·73-s + 0.341·77-s + 0.112·79-s − 0.105·89-s − 0.314·91-s − 0.304·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.66138978302396, −14.34250732214120, −14.00534979586318, −13.37785140011984, −12.76734688121508, −12.16885426471188, −11.80949041258687, −11.17362048265853, −10.86479391417908, −10.18874841343944, −9.653467451851714, −9.013302394107145, −8.536752150758392, −7.996561305656979, −7.492205922995593, −6.884749595683941, −6.320123383542365, −5.587572079570738, −5.028558769765999, −4.579352400961878, −3.853543670828908, −3.296583820660909, −2.240891757631495, −1.894439984453808, −1.049882744461310, 0,
1.049882744461310, 1.894439984453808, 2.240891757631495, 3.296583820660909, 3.853543670828908, 4.579352400961878, 5.028558769765999, 5.587572079570738, 6.320123383542365, 6.884749595683941, 7.492205922995593, 7.996561305656979, 8.536752150758392, 9.013302394107145, 9.653467451851714, 10.18874841343944, 10.86479391417908, 11.17362048265853, 11.80949041258687, 12.16885426471188, 12.76734688121508, 13.37785140011984, 14.00534979586318, 14.34250732214120, 14.66138978302396
