| L(s) = 1 | + 7-s − 2·17-s + 2·19-s − 8·23-s − 2·29-s − 4·31-s + 6·37-s + 2·41-s + 8·43-s + 4·47-s + 49-s − 10·53-s + 6·59-s + 4·61-s − 12·67-s + 14·73-s + 8·79-s − 6·83-s − 10·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.485·17-s + 0.458·19-s − 1.66·23-s − 0.371·29-s − 0.718·31-s + 0.986·37-s + 0.312·41-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 1.37·53-s + 0.781·59-s + 0.512·61-s − 1.46·67-s + 1.63·73-s + 0.900·79-s − 0.658·83-s − 1.05·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.58831940576567, −15.13624938859189, −14.47028842512725, −14.04099629370975, −13.61697578337544, −12.85372424599774, −12.44436424730137, −11.84742401102871, −11.20248397033093, −10.90919657785230, −10.13499930427312, −9.580587839964538, −9.115867806051769, −8.381940310321275, −7.832259530290109, −7.397779836412276, −6.641860037131184, −5.932516711621735, −5.545833590522006, −4.668502697145597, −4.134174510854198, −3.513410404647816, −2.545514626949905, −1.991129623802998, −1.075367918582860, 0,
1.075367918582860, 1.991129623802998, 2.545514626949905, 3.513410404647816, 4.134174510854198, 4.668502697145597, 5.545833590522006, 5.932516711621735, 6.641860037131184, 7.397779836412276, 7.832259530290109, 8.381940310321275, 9.115867806051769, 9.580587839964538, 10.13499930427312, 10.90919657785230, 11.20248397033093, 11.84742401102871, 12.44436424730137, 12.85372424599774, 13.61697578337544, 14.04099629370975, 14.47028842512725, 15.13624938859189, 15.58831940576567
