L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)7-s + (−0.623 − 0.781i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + (−0.277 − 1.21i)11-s + (−0.400 − 1.75i)13-s + (−0.900 − 0.433i)14-s + (0.623 − 0.781i)16-s − 18-s + 1.24·19-s + (0.900 + 0.433i)20-s + (1.12 − 0.541i)22-s + (1.80 − 0.867i)23-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.623 + 0.781i)7-s + (−0.623 − 0.781i)8-s + (−0.222 + 0.974i)9-s + (0.623 − 0.781i)10-s + (−0.277 − 1.21i)11-s + (−0.400 − 1.75i)13-s + (−0.900 − 0.433i)14-s + (0.623 − 0.781i)16-s − 18-s + 1.24·19-s + (0.900 + 0.433i)20-s + (1.12 − 0.541i)22-s + (1.80 − 0.867i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7486273391\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7486273391\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (0.623 + 0.781i)T \) |
| 7 | \( 1 + (0.623 - 0.781i)T \) |
good | 3 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 + (-1.80 + 0.867i)T + (0.623 - 0.781i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 41 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 61 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.919210961427262705465857864120, −8.460555940354344019367530097644, −7.86180188050812674664054151638, −7.11943376578479068331239684037, −5.94117863130637479914192307419, −5.18244004807877483170287403004, −5.04674568378654813228767786025, −3.39913469494894259084939700350, −2.94457408208716151089085442966, −0.55416397417218741651832472477,
1.36799964246172607348029125150, 2.79185358168885177684246247060, 3.46784887358781317162931385852, 4.26042808097085803478767183944, 5.02404279972221394845414603590, 6.44307878901374921388141182907, 6.99517101733597370417443810347, 7.68440490481201449590650190165, 9.096416713624936518478420741362, 9.544598224219223698436130666966