L(s) = 1 | + 3-s − 2·4-s − 5-s − 7-s + 9-s + 3·11-s − 2·12-s − 15-s + 4·16-s − 17-s + 5·19-s + 2·20-s − 21-s − 4·23-s + 25-s + 27-s + 2·28-s + 3·29-s + 8·31-s + 3·33-s + 35-s − 2·36-s + 4·37-s + 9·41-s − 11·43-s − 6·44-s − 45-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.577·12-s − 0.258·15-s + 16-s − 0.242·17-s + 1.14·19-s + 0.447·20-s − 0.218·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.557·29-s + 1.43·31-s + 0.522·33-s + 0.169·35-s − 1/3·36-s + 0.657·37-s + 1.40·41-s − 1.67·43-s − 0.904·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.010713773\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.010713773\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.81651037679258, −12.23363660826887, −11.76242679886862, −11.56557353413332, −10.74444167113557, −10.23700793063492, −9.733168920073561, −9.519724433565101, −9.020560430911089, −8.562182713103533, −7.982454995698946, −7.859283058102709, −7.143817077535708, −6.569476068690726, −6.151793290609539, −5.598288110002976, −4.797213287179059, −4.573377054022079, −3.996627471289824, −3.550906691032306, −3.046735330448970, −2.559437675257598, −1.567785836688363, −1.108731763133306, −0.4148107038879765,
0.4148107038879765, 1.108731763133306, 1.567785836688363, 2.559437675257598, 3.046735330448970, 3.550906691032306, 3.996627471289824, 4.573377054022079, 4.797213287179059, 5.598288110002976, 6.151793290609539, 6.569476068690726, 7.143817077535708, 7.859283058102709, 7.982454995698946, 8.562182713103533, 9.020560430911089, 9.519724433565101, 9.733168920073561, 10.23700793063492, 10.74444167113557, 11.56557353413332, 11.76242679886862, 12.23363660826887, 12.81651037679258