| L(s) = 1 | − 3.25i·3-s + (−2.54 − 2.54i)7-s − 7.61·9-s + (0.462 + 0.462i)11-s + 1.33·13-s + (−2.37 − 2.37i)17-s + (−2.69 − 2.69i)19-s + (−8.27 + 8.27i)21-s + (2.10 − 2.10i)23-s + 15.0i·27-s + (−1.97 + 1.97i)29-s + 7.03i·31-s + (1.50 − 1.50i)33-s + 7.81·37-s − 4.34i·39-s + ⋯ |
| L(s) = 1 | − 1.88i·3-s + (−0.960 − 0.960i)7-s − 2.53·9-s + (0.139 + 0.139i)11-s + 0.370·13-s + (−0.575 − 0.575i)17-s + (−0.618 − 0.618i)19-s + (−1.80 + 1.80i)21-s + (0.438 − 0.438i)23-s + 2.89i·27-s + (−0.367 + 0.367i)29-s + 1.26i·31-s + (0.262 − 0.262i)33-s + 1.28·37-s − 0.696i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5457945637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5457945637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.25iT - 3T^{2} \) |
| 7 | \( 1 + (2.54 + 2.54i)T + 7iT^{2} \) |
| 11 | \( 1 + (-0.462 - 0.462i)T + 11iT^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 + (2.37 + 2.37i)T + 17iT^{2} \) |
| 19 | \( 1 + (2.69 + 2.69i)T + 19iT^{2} \) |
| 23 | \( 1 + (-2.10 + 2.10i)T - 23iT^{2} \) |
| 29 | \( 1 + (1.97 - 1.97i)T - 29iT^{2} \) |
| 31 | \( 1 - 7.03iT - 31T^{2} \) |
| 37 | \( 1 - 7.81T + 37T^{2} \) |
| 41 | \( 1 + 2.17iT - 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 + (-0.0727 + 0.0727i)T - 47iT^{2} \) |
| 53 | \( 1 - 0.719iT - 53T^{2} \) |
| 59 | \( 1 + (8.67 - 8.67i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.10 + 7.10i)T + 61iT^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + (0.905 + 0.905i)T + 73iT^{2} \) |
| 79 | \( 1 - 3.90T + 79T^{2} \) |
| 83 | \( 1 - 6.02iT - 83T^{2} \) |
| 89 | \( 1 - 7.46T + 89T^{2} \) |
| 97 | \( 1 + (3.74 + 3.74i)T + 97iT^{2} \) |
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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
−8.723501747568548654967081558915, −7.81605232336763230034187606516, −7.02241384084483042245290344629, −6.69731832392649328555752422711, −5.98568046711156598874157174716, −4.71000624765918746052212043595, −3.37358828708584944283651186991, −2.50411937609144538577726730413, −1.27512987312468938707571556255, −0.21622397258947829991655192460,
2.40322983996214667925124698508, 3.36524612308895275896059855651, 4.04345198719909971707717389283, 4.92794012147838271594326467692, 6.05918072670142697515091304578, 6.14279849656393344842406370091, 7.924412770606259963839537167025, 8.742403546077420439397856704024, 9.352203297913709259966964407427, 9.770537111513472815549114324302
