| L(s) = 1 | − 2·5-s − 7-s − 3·9-s − 6·13-s + 2·17-s − 8·19-s − 25-s − 29-s + 4·31-s + 2·35-s − 2·37-s + 2·41-s − 4·43-s + 6·45-s + 4·47-s + 49-s − 10·53-s − 12·59-s − 2·61-s + 3·63-s + 12·65-s − 12·67-s − 8·71-s + 10·73-s + 9·81-s + 12·83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 9-s − 1.66·13-s + 0.485·17-s − 1.83·19-s − 1/5·25-s − 0.185·29-s + 0.718·31-s + 0.338·35-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 0.894·45-s + 0.583·47-s + 1/7·49-s − 1.37·53-s − 1.56·59-s − 0.256·61-s + 0.377·63-s + 1.48·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s + 81-s + 1.31·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 393008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−12.41949818027457, −12.24767480962538, −11.89838977970345, −11.40107333449313, −10.78108603198561, −10.56982730637550, −10.00687474967706, −9.370280099990006, −9.184218828317172, −8.402641135086704, −8.216108899419602, −7.556021749486302, −7.415108799088690, −6.623109941989417, −6.236224511700855, −5.839888933611601, −5.072259543411209, −4.712351934398425, −4.280178621708488, −3.602451082149358, −3.185732357054625, −2.557209787002719, −2.221288439716974, −1.409166355053657, −0.3663301893751763, 0,
0.3663301893751763, 1.409166355053657, 2.221288439716974, 2.557209787002719, 3.185732357054625, 3.602451082149358, 4.280178621708488, 4.712351934398425, 5.072259543411209, 5.839888933611601, 6.236224511700855, 6.623109941989417, 7.415108799088690, 7.556021749486302, 8.216108899419602, 8.402641135086704, 9.184218828317172, 9.370280099990006, 10.00687474967706, 10.56982730637550, 10.78108603198561, 11.40107333449313, 11.89838977970345, 12.24767480962538, 12.41949818027457
