| L(s) = 1 | + 3.17i·3-s − 4.23i·5-s − 7-s − 7.08·9-s + 4.99i·11-s + 0.439i·13-s + 13.4·15-s + 3.50·17-s − 3.45i·19-s − 3.17i·21-s + 4.39·23-s − 12.9·25-s − 12.9i·27-s − 0.132i·29-s − 5.55·31-s + ⋯ |
| L(s) = 1 | + 1.83i·3-s − 1.89i·5-s − 0.377·7-s − 2.36·9-s + 1.50i·11-s + 0.121i·13-s + 3.47·15-s + 0.849·17-s − 0.791i·19-s − 0.693i·21-s + 0.916·23-s − 2.59·25-s − 2.49i·27-s − 0.0246i·29-s − 0.998·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4807723164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4807723164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.17iT - 3T^{2} \) |
| 5 | \( 1 + 4.23iT - 5T^{2} \) |
| 11 | \( 1 - 4.99iT - 11T^{2} \) |
| 13 | \( 1 - 0.439iT - 13T^{2} \) |
| 17 | \( 1 - 3.50T + 17T^{2} \) |
| 19 | \( 1 + 3.45iT - 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 + 0.132iT - 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 - 0.572iT - 37T^{2} \) |
| 41 | \( 1 + 5.42T + 41T^{2} \) |
| 43 | \( 1 + 5.69iT - 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 3.65iT - 53T^{2} \) |
| 59 | \( 1 + 2.20iT - 59T^{2} \) |
| 61 | \( 1 + 9.89iT - 61T^{2} \) |
| 67 | \( 1 - 8.51iT - 67T^{2} \) |
| 71 | \( 1 + 3.11T + 71T^{2} \) |
| 73 | \( 1 + 9.87T + 73T^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 + 2.48iT - 83T^{2} \) |
| 89 | \( 1 - 5.06T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−8.723713038070912261300366020919, −7.937180093729662850216763616211, −6.88416381159167810165939796551, −5.55408036623665087951106483913, −5.17311409978444672698203015311, −4.56870332625325282573880147565, −4.02956038466651841468216241368, −3.07105775720577733255336761160, −1.68375713647742606423178726524, −0.14565337735352476439112127715,
1.22622436765739580103464115179, 2.31901638698335075140883721564, 3.26357420537428731065020064245, 3.36031682516794542386855318058, 5.53736784893056065401485704010, 6.02689742595516934833884732278, 6.61905337795194929539341264189, 7.15660989616702301458118787664, 7.83986093501594066321695974392, 8.334028278582121421602228288033
