| L(s) = 1 | + (0.809 − 0.587i)2-s + (0.241 − 0.743i)3-s + (0.309 − 0.951i)4-s + (1.92 − 1.14i)5-s + (−0.241 − 0.743i)6-s − 7-s + (−0.309 − 0.951i)8-s + (1.93 + 1.40i)9-s + (0.885 − 2.05i)10-s + (0.229 − 0.166i)11-s + (−0.632 − 0.459i)12-s + (−1.15 − 0.836i)13-s + (−0.809 + 0.587i)14-s + (−0.383 − 1.70i)15-s + (−0.809 − 0.587i)16-s + (−0.450 − 1.38i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.139 − 0.429i)3-s + (0.154 − 0.475i)4-s + (0.860 − 0.510i)5-s + (−0.0986 − 0.303i)6-s − 0.377·7-s + (−0.109 − 0.336i)8-s + (0.644 + 0.468i)9-s + (0.280 − 0.649i)10-s + (0.0692 − 0.0502i)11-s + (−0.182 − 0.132i)12-s + (−0.319 − 0.232i)13-s + (−0.216 + 0.157i)14-s + (−0.0989 − 0.440i)15-s + (−0.202 − 0.146i)16-s + (−0.109 − 0.336i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65157 - 1.27778i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65157 - 1.27778i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-1.92 + 1.14i)T \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + (-0.241 + 0.743i)T + (-2.42 - 1.76i)T^{2} \) |
| 11 | \( 1 + (-0.229 + 0.166i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.15 + 0.836i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.450 + 1.38i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.0630 + 0.194i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (7.07 - 5.13i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.161 + 0.498i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.37 - 4.24i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.49 - 4.71i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.08 - 0.790i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 + (-1.27 + 3.92i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (2.12 - 6.54i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.96 - 2.87i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.48 + 1.07i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.69 - 5.21i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (1.13 - 3.47i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (11.6 - 8.45i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (5.20 - 16.0i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.64 + 5.07i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-8.07 + 5.86i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.81 + 14.8i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.51524641360594149049961631839, −10.12725223832743335138132795261, −9.838039403329353605456504845263, −8.550564708620605019591428043168, −7.37065728403787189934316167770, −6.29457479021768656858712409118, −5.33423597602577812553307361008, −4.25935279817651162805626164086, −2.67954215323168162518530684817, −1.48115898942119609256960287851,
2.26084145896832469221021942119, 3.63622073433343928747130297556, 4.65475622113656370242709457562, 6.03637258500967704960802934075, 6.59252714721107029405547563772, 7.72782106980005365199461899411, 9.055910396146717385676438452460, 9.873514282681851958653193974818, 10.56849652787805956865832384949, 11.83520528750946748854228794971
