L(s) = 1 | + (0.955 − 0.294i)3-s + (−0.0747 + 0.997i)7-s + (0.826 − 0.563i)9-s + (0.255 + 0.531i)13-s + (0.0747 − 0.129i)19-s + (0.222 + 0.974i)21-s + (−0.0747 − 0.997i)25-s + (0.623 − 0.781i)27-s + (0.988 + 1.71i)31-s + (0.722 − 0.108i)37-s + (0.400 + 0.432i)39-s + (−1.09 + 0.250i)43-s + (−0.988 − 0.149i)49-s + (0.0332 − 0.145i)57-s + (−0.233 − 1.54i)61-s + ⋯ |
L(s) = 1 | + (0.955 − 0.294i)3-s + (−0.0747 + 0.997i)7-s + (0.826 − 0.563i)9-s + (0.255 + 0.531i)13-s + (0.0747 − 0.129i)19-s + (0.222 + 0.974i)21-s + (−0.0747 − 0.997i)25-s + (0.623 − 0.781i)27-s + (0.988 + 1.71i)31-s + (0.722 − 0.108i)37-s + (0.400 + 0.432i)39-s + (−1.09 + 0.250i)43-s + (−0.988 − 0.149i)49-s + (0.0332 − 0.145i)57-s + (−0.233 − 1.54i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.706967864\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.706967864\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.955 + 0.294i)T \) |
| 7 | \( 1 + (0.0747 - 0.997i)T \) |
good | 5 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 11 | \( 1 + (0.365 + 0.930i)T^{2} \) |
| 13 | \( 1 + (-0.255 - 0.531i)T + (-0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + (-0.733 - 0.680i)T^{2} \) |
| 19 | \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.733 + 0.680i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.722 + 0.108i)T + (0.955 - 0.294i)T^{2} \) |
| 41 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (1.09 - 0.250i)T + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 53 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 59 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 61 | \( 1 + (0.233 + 1.54i)T + (-0.955 + 0.294i)T^{2} \) |
| 67 | \( 1 + (1.17 - 0.680i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-1.85 + 0.139i)T + (0.988 - 0.149i)T^{2} \) |
| 79 | \( 1 + (0.258 + 0.149i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (0.365 - 0.930i)T^{2} \) |
| 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
−9.085178894855516449239439719141, −8.412901428950399406448322205441, −7.925847958964666486851844507077, −6.75088530146806863212635657622, −6.36907379998295425058051536136, −5.17847506473919038882922187712, −4.30392902700946112367667590703, −3.23566745266946186648757365311, −2.50973342558212575207553800728, −1.50637813220103621201778098390,
1.25300934939407960684611742031, 2.53848156751059836962600290622, 3.48576569688225305305302778220, 4.13806038553738945329885435615, 5.00108465372538484471294723053, 6.10499175913519489503932452085, 7.05899852314734237604968973333, 7.76921043674197711556366848522, 8.240068068634462539837126180966, 9.244596866226634031895183929949