| L(s) = 1 | − 7-s − 3·9-s − 11-s − 6·13-s − 2·19-s + 4·23-s − 5·25-s − 2·29-s + 2·31-s + 2·37-s − 8·41-s − 2·47-s + 49-s − 10·53-s − 4·59-s + 10·61-s + 3·63-s + 4·67-s − 8·73-s + 77-s + 8·79-s + 9·81-s − 2·83-s − 6·89-s + 6·91-s + 2·97-s + 3·99-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.301·11-s − 1.66·13-s − 0.458·19-s + 0.834·23-s − 25-s − 0.371·29-s + 0.359·31-s + 0.328·37-s − 1.24·41-s − 0.291·47-s + 1/7·49-s − 1.37·53-s − 0.520·59-s + 1.28·61-s + 0.377·63-s + 0.488·67-s − 0.936·73-s + 0.113·77-s + 0.900·79-s + 81-s − 0.219·83-s − 0.635·89-s + 0.628·91-s + 0.203·97-s + 0.301·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 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
−10.07995037027806350781622361938, −9.427364167513449399975184048294, −8.427472592413009369442981171559, −7.56186589309529403797122372766, −6.60336419837693223802202901848, −5.55808150974400203762583292939, −4.69735623090568767734453095829, −3.26677825137831447402592197381, −2.27748515390590495115985369379, 0,
2.27748515390590495115985369379, 3.26677825137831447402592197381, 4.69735623090568767734453095829, 5.55808150974400203762583292939, 6.60336419837693223802202901848, 7.56186589309529403797122372766, 8.427472592413009369442981171559, 9.427364167513449399975184048294, 10.07995037027806350781622361938
