| L(s) = 1 | − i·3-s + 0.248·5-s + (0.652 − 2.56i)7-s − 9-s + 4.05i·11-s + 5.46i·13-s − 0.248i·15-s − 0.349·17-s − 3.87·19-s + (−2.56 − 0.652i)21-s + (0.300 + 4.78i)23-s − 4.93·25-s + i·27-s − 3.50·29-s + 7.87i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.110·5-s + (0.246 − 0.969i)7-s − 0.333·9-s + 1.22i·11-s + 1.51i·13-s − 0.0640i·15-s − 0.0848·17-s − 0.890·19-s + (−0.559 − 0.142i)21-s + (0.0626 + 0.998i)23-s − 0.987·25-s + 0.192i·27-s − 0.651·29-s + 1.41i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079130566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079130566\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + (-0.652 + 2.56i)T \) |
| 23 | \( 1 + (-0.300 - 4.78i)T \) |
| good | 5 | \( 1 - 0.248T + 5T^{2} \) |
| 11 | \( 1 - 4.05iT - 11T^{2} \) |
| 13 | \( 1 - 5.46iT - 13T^{2} \) |
| 17 | \( 1 + 0.349T + 17T^{2} \) |
| 19 | \( 1 + 3.87T + 19T^{2} \) |
| 29 | \( 1 + 3.50T + 29T^{2} \) |
| 31 | \( 1 - 7.87iT - 31T^{2} \) |
| 37 | \( 1 - 3.76iT - 37T^{2} \) |
| 41 | \( 1 - 5.40iT - 41T^{2} \) |
| 43 | \( 1 + 11.4iT - 43T^{2} \) |
| 47 | \( 1 - 1.86iT - 47T^{2} \) |
| 53 | \( 1 + 9.71iT - 53T^{2} \) |
| 59 | \( 1 - 12.5iT - 59T^{2} \) |
| 61 | \( 1 + 4.03T + 61T^{2} \) |
| 67 | \( 1 + 4.70iT - 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 9.64iT - 73T^{2} \) |
| 79 | \( 1 - 6.98iT - 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 6.84T + 89T^{2} \) |
| 97 | \( 1 + 1.02T + 97T^{2} \) |
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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
−9.419657804080942728411380108504, −8.528551782064651621148491260501, −7.62969011096873416798925370557, −6.97557338112863750280349296236, −6.54328363997679523252967205240, −5.28894507958872673795949085419, −4.40038395630791515269543732255, −3.69079369470512211041707215256, −2.12233614949580498014117232862, −1.49510708119052732597117082630,
0.37615416014225422172682421639, 2.21134736294767866086023826035, 3.06393502804701492231393789719, 4.02028497939806504047713464263, 5.09769277528452087164456153240, 5.85235657252270201536038361368, 6.25175538284014465625988275518, 7.85223679380435728844368568840, 8.209561653541378756014262983479, 9.052731520711909011876941993659
