| L(s) = 1 | − 2-s + 4-s + 5·7-s − 8-s + 3·11-s + 13-s − 5·14-s + 16-s − 3·17-s − 4·19-s − 3·22-s − 6·23-s − 26-s + 5·28-s − 9·29-s + 5·31-s − 32-s + 3·34-s + 2·37-s + 4·38-s + 2·43-s + 3·44-s + 6·46-s + 9·47-s + 18·49-s + 52-s + 9·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.88·7-s − 0.353·8-s + 0.904·11-s + 0.277·13-s − 1.33·14-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 0.639·22-s − 1.25·23-s − 0.196·26-s + 0.944·28-s − 1.67·29-s + 0.898·31-s − 0.176·32-s + 0.514·34-s + 0.328·37-s + 0.648·38-s + 0.304·43-s + 0.452·44-s + 0.884·46-s + 1.31·47-s + 18/7·49-s + 0.138·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.867092315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.867092315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.176997885974374628561748067806, −7.59418826518993917969060308454, −6.82962101168113154682468153413, −6.03067242691010449186717723785, −5.31435733226551836326059702544, −4.27550335332108452765120105760, −3.92089617066885854760092218201, −2.28130999511561120477173054203, −1.87270370507269017712517494275, −0.831442248746084151852258105409,
0.831442248746084151852258105409, 1.87270370507269017712517494275, 2.28130999511561120477173054203, 3.92089617066885854760092218201, 4.27550335332108452765120105760, 5.31435733226551836326059702544, 6.03067242691010449186717723785, 6.82962101168113154682468153413, 7.59418826518993917969060308454, 8.176997885974374628561748067806
