| L(s) = 1 | + 2·7-s − 13-s − 6·17-s − 2·19-s + 6·29-s − 2·31-s − 2·37-s + 12·41-s − 4·43-s − 3·49-s + 6·53-s + 12·59-s + 2·61-s − 10·67-s + 12·71-s − 14·73-s − 8·79-s − 12·83-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 12·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.277·13-s − 1.45·17-s − 0.458·19-s + 1.11·29-s − 0.359·31-s − 0.328·37-s + 1.87·41-s − 0.609·43-s − 3/7·49-s + 0.824·53-s + 1.56·59-s + 0.256·61-s − 1.22·67-s + 1.42·71-s − 1.63·73-s − 0.900·79-s − 1.31·83-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.10·119-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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.78923407996663, −14.37863213363342, −13.92278905069316, −13.21280154791293, −12.91520928438099, −12.28775145477028, −11.60156741606193, −11.34763533749984, −10.70349238907862, −10.29628475219643, −9.623248468376911, −8.994674258650410, −8.506671516253859, −8.147612855941745, −7.320217909116739, −6.944712813904504, −6.301395773847251, −5.665359055361538, −5.032182268720344, −4.358466907476111, −4.136983891397495, −3.075241239939885, −2.418532980419055, −1.864644701150641, −0.9859611574818484, 0,
0.9859611574818484, 1.864644701150641, 2.418532980419055, 3.075241239939885, 4.136983891397495, 4.358466907476111, 5.032182268720344, 5.665359055361538, 6.301395773847251, 6.944712813904504, 7.320217909116739, 8.147612855941745, 8.506671516253859, 8.994674258650410, 9.623248468376911, 10.29628475219643, 10.70349238907862, 11.34763533749984, 11.60156741606193, 12.28775145477028, 12.91520928438099, 13.21280154791293, 13.92278905069316, 14.37863213363342, 14.78923407996663
