L(s) = 1 | + (0.541 − 1.66i)3-s + (0.309 + 0.951i)4-s + (1.50 − 1.09i)5-s + (0.331 + 1.01i)7-s + (−1.67 − 1.21i)9-s + (−0.187 + 0.982i)11-s + 1.75·12-s + (−1.17 − 0.856i)13-s + (−1.00 − 3.09i)15-s + (−0.809 + 0.587i)16-s + (1.50 + 1.09i)20-s + 1.87·21-s + (0.759 − 2.33i)25-s + (−1.51 + 1.10i)27-s + (−0.866 + 0.629i)28-s + ⋯ |
L(s) = 1 | + (0.541 − 1.66i)3-s + (0.309 + 0.951i)4-s + (1.50 − 1.09i)5-s + (0.331 + 1.01i)7-s + (−1.67 − 1.21i)9-s + (−0.187 + 0.982i)11-s + 1.75·12-s + (−1.17 − 0.856i)13-s + (−1.00 − 3.09i)15-s + (−0.809 + 0.587i)16-s + (1.50 + 1.09i)20-s + 1.87·21-s + (0.759 − 2.33i)25-s + (−1.51 + 1.10i)27-s + (−0.866 + 0.629i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.668696702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.668696702\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.187 - 0.982i)T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-1.50 + 1.09i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.331 - 1.01i)T + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (1.17 + 0.856i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.27T + T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.598 - 1.84i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-1.50 + 1.09i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−9.371345011806597915494488993907, −8.436267539562733958427163119063, −8.175996030978487469734371418785, −7.21253620110141079416704797676, −6.48407572596504224048809473524, −5.54536580723250139552932338256, −4.80703768568097760864930159614, −2.83792465212348602887162471077, −2.28482034253142266103178364332, −1.59040690498559854027892320784,
1.96735912859774328563135863310, 2.82337589111669778181486003975, 3.86119133159626993095609393741, 5.00261733748109193600448407009, 5.50034725476555119192310463714, 6.53284635600522783307680798625, 7.24387849318737470156499799414, 8.647529141222961817914262946548, 9.502326350821678508280091807148, 9.938292430671806831416739685070