| L(s) = 1 | − 2·5-s + 7-s + 3·11-s + 17-s − 5·19-s − 2·23-s − 25-s − 29-s + 2·31-s − 2·35-s − 2·37-s + 5·41-s + 2·43-s − 3·47-s + 49-s + 9·53-s − 6·55-s − 6·59-s + 13·61-s − 2·67-s − 12·71-s + 8·73-s + 3·77-s − 17·79-s − 2·85-s − 5·89-s + 10·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.904·11-s + 0.242·17-s − 1.14·19-s − 0.417·23-s − 1/5·25-s − 0.185·29-s + 0.359·31-s − 0.338·35-s − 0.328·37-s + 0.780·41-s + 0.304·43-s − 0.437·47-s + 1/7·49-s + 1.23·53-s − 0.809·55-s − 0.781·59-s + 1.66·61-s − 0.244·67-s − 1.42·71-s + 0.936·73-s + 0.341·77-s − 1.91·79-s − 0.216·85-s − 0.529·89-s + 1.02·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 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
−14.27442340250010, −13.76292534202734, −13.09053582221821, −12.67787333496838, −11.98982944621207, −11.82671477638410, −11.26037318442240, −10.80986170410600, −10.20749757102643, −9.721938430358512, −9.020296909688335, −8.617362514787215, −8.112377996916491, −7.661022181496977, −7.023960799382872, −6.623494258542199, −5.886716628501138, −5.488616037860862, −4.520515413911813, −4.244595415478906, −3.781335217156548, −3.081704929681520, −2.289414796360923, −1.647254506349772, −0.8463342136399954, 0,
0.8463342136399954, 1.647254506349772, 2.289414796360923, 3.081704929681520, 3.781335217156548, 4.244595415478906, 4.520515413911813, 5.488616037860862, 5.886716628501138, 6.623494258542199, 7.023960799382872, 7.661022181496977, 8.112377996916491, 8.617362514787215, 9.020296909688335, 9.721938430358512, 10.20749757102643, 10.80986170410600, 11.26037318442240, 11.82671477638410, 11.98982944621207, 12.67787333496838, 13.09053582221821, 13.76292534202734, 14.27442340250010
