L(s) = 1 | + 4·5-s + 7-s − 2·13-s + 2·17-s + 6·19-s − 2·23-s + 11·25-s − 6·29-s + 8·31-s + 4·35-s + 6·37-s + 6·41-s + 10·43-s + 2·47-s + 49-s + 12·53-s + 4·59-s + 6·61-s − 8·65-s + 4·67-s − 6·71-s + 16·73-s − 4·79-s − 4·83-s + 8·85-s − 18·89-s − 2·91-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 0.377·7-s − 0.554·13-s + 0.485·17-s + 1.37·19-s − 0.417·23-s + 11/5·25-s − 1.11·29-s + 1.43·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s + 1.52·43-s + 0.291·47-s + 1/7·49-s + 1.64·53-s + 0.520·59-s + 0.768·61-s − 0.992·65-s + 0.488·67-s − 0.712·71-s + 1.87·73-s − 0.450·79-s − 0.439·83-s + 0.867·85-s − 1.90·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.659845227\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.659845227\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.74146966776522, −12.99870716786411, −12.78601502751526, −12.13589800205536, −11.58349874013483, −11.16160048916165, −10.47518887785753, −10.04150444823580, −9.661871053613781, −9.354318555367782, −8.775747502973087, −8.129574302303948, −7.533910057003941, −7.137341800488107, −6.476255530922627, −5.793976977494847, −5.628188704015646, −5.137978450072636, −4.449213101847061, −3.847154644688314, −2.939472377131984, −2.470226813180463, −2.062291497093286, −1.155981536346242, −0.8206027936166356,
0.8206027936166356, 1.155981536346242, 2.062291497093286, 2.470226813180463, 2.939472377131984, 3.847154644688314, 4.449213101847061, 5.137978450072636, 5.628188704015646, 5.793976977494847, 6.476255530922627, 7.137341800488107, 7.533910057003941, 8.129574302303948, 8.775747502973087, 9.354318555367782, 9.661871053613781, 10.04150444823580, 10.47518887785753, 11.16160048916165, 11.58349874013483, 12.13589800205536, 12.78601502751526, 12.99870716786411, 13.74146966776522