| L(s) = 1 | − i·3-s − 2.24·5-s + (−0.274 − 2.63i)7-s − 9-s − 5.61i·11-s − 3.17i·13-s + 2.24i·15-s + 0.653·17-s − 1.73·19-s + (−2.63 + 0.274i)21-s + (3.98 + 2.66i)23-s + 0.0182·25-s + i·27-s + 0.830·29-s + 3.41i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.00·5-s + (−0.103 − 0.994i)7-s − 0.333·9-s − 1.69i·11-s − 0.881i·13-s + 0.578i·15-s + 0.158·17-s − 0.398·19-s + (−0.574 + 0.0599i)21-s + (0.831 + 0.555i)23-s + 0.00365·25-s + 0.192i·27-s + 0.154·29-s + 0.613i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6333617645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6333617645\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.274 + 2.63i)T \) |
| 23 | \( 1 + (-3.98 - 2.66i)T \) |
| good | 5 | \( 1 + 2.24T + 5T^{2} \) |
| 11 | \( 1 + 5.61iT - 11T^{2} \) |
| 13 | \( 1 + 3.17iT - 13T^{2} \) |
| 17 | \( 1 - 0.653T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 29 | \( 1 - 0.830T + 29T^{2} \) |
| 31 | \( 1 - 3.41iT - 31T^{2} \) |
| 37 | \( 1 + 9.03iT - 37T^{2} \) |
| 41 | \( 1 - 4.58iT - 41T^{2} \) |
| 43 | \( 1 - 1.46iT - 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 0.509iT - 53T^{2} \) |
| 59 | \( 1 + 1.43iT - 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 4.99iT - 67T^{2} \) |
| 71 | \( 1 + 3.32T + 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 3.98iT - 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 5.52T + 89T^{2} \) |
| 97 | \( 1 - 2.63T + 97T^{2} \) |
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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
−8.519166180229247016680436852328, −7.82142225110540225954723423355, −7.42997181621926961289317183605, −6.40476313608819735870201988335, −5.67553730596942979893205304093, −4.54710018957084794304352989856, −3.51150002833962080229718597392, −3.02874233141340939938662942014, −1.18270496599599488698434228874, −0.25393578881665515527816801401,
1.90270190878599739115461861653, 2.91065009657425339857454400233, 4.10263347186420494971903392811, 4.58703899776614945477069174818, 5.47775048601988599354561095717, 6.63603683754110239140209174485, 7.24367994291449911091865821477, 8.247276964560561586924940571862, 8.853558369696614953016185583743, 9.639069426996905168672725163199
