L(s) = 1 | + (0.222 − 0.974i)3-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)12-s + (0.222 + 0.974i)13-s + (−0.222 − 0.974i)16-s + 0.867i·19-s + (0.222 + 0.974i)21-s + (0.900 + 0.433i)25-s + (−0.623 + 0.781i)27-s + (0.222 − 0.974i)28-s + 1.56i·31-s + (0.900 − 0.433i)36-s + (−1.22 + 0.974i)37-s + ⋯ |
L(s) = 1 | + (0.222 − 0.974i)3-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)7-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)12-s + (0.222 + 0.974i)13-s + (−0.222 − 0.974i)16-s + 0.867i·19-s + (0.222 + 0.974i)21-s + (0.900 + 0.433i)25-s + (−0.623 + 0.781i)27-s + (0.222 − 0.974i)28-s + 1.56i·31-s + (0.900 − 0.433i)36-s + (−1.22 + 0.974i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7078436431\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7078436431\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.222 + 0.974i)T \) |
| 7 | \( 1 + (0.900 - 0.433i)T \) |
| 13 | \( 1 + (-0.222 - 0.974i)T \) |
good | 2 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 5 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 11 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 19 | \( 1 - 0.867iT - T^{2} \) |
| 23 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 - 1.56iT - T^{2} \) |
| 37 | \( 1 + (1.22 - 0.974i)T + (0.222 - 0.974i)T^{2} \) |
| 41 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 47 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 53 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 61 | \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 67 | \( 1 - 1.94iT - T^{2} \) |
| 71 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.678 + 1.40i)T + (-0.623 - 0.781i)T^{2} \) |
| 79 | \( 1 + 1.24T + T^{2} \) |
| 83 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 + 0.867iT - T^{2} \) |
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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
−9.235562248413805981504682517499, −8.736545866439784829212463584223, −8.113376120606596084347978925418, −7.10797671251881982200861291433, −6.64080123554589201993123860109, −5.69664216271924606355725246671, −4.62024973683610817110241332834, −3.43600009006879558460643817162, −2.92018213559089562099993325316, −1.54452901296351434954615398343,
0.52015482043541351460326387282, 2.48644845506856831680176616664, 3.55107193131529869487269428697, 4.25663410994923618895804826789, 5.20581480491753057806433560051, 5.81028471259674975943899546579, 6.73937679846215164851542708970, 7.82560581529999398345311329571, 8.850001229405004371983541913245, 9.215880669452303808397119005360