| L(s) = 1 | − 3-s + i·5-s + (2.46 + 0.953i)7-s + 9-s + 4.70i·11-s + 2.93i·13-s − i·15-s + 3.31i·17-s − 4.77·19-s + (−2.46 − 0.953i)21-s − 6.24i·23-s − 25-s − 27-s + 10.0·29-s − 10.0·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447i·5-s + (0.932 + 0.360i)7-s + 0.333·9-s + 1.41i·11-s + 0.814i·13-s − 0.258i·15-s + 0.803i·17-s − 1.09·19-s + (−0.538 − 0.208i)21-s − 1.30i·23-s − 0.200·25-s − 0.192·27-s + 1.87·29-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.115459851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.115459851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.46 - 0.953i)T \) |
| good | 11 | \( 1 - 4.70iT - 11T^{2} \) |
| 13 | \( 1 - 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 3.31iT - 17T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 + 6.24iT - 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 + 9.92iT - 41T^{2} \) |
| 43 | \( 1 - 9.55iT - 43T^{2} \) |
| 47 | \( 1 + 7.14T + 47T^{2} \) |
| 53 | \( 1 - 5.21T + 53T^{2} \) |
| 59 | \( 1 - 4.31T + 59T^{2} \) |
| 61 | \( 1 - 7.53iT - 61T^{2} \) |
| 67 | \( 1 + 6.48iT - 67T^{2} \) |
| 71 | \( 1 + 1.65iT - 71T^{2} \) |
| 73 | \( 1 + 2.06iT - 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 - 13.0iT - 89T^{2} \) |
| 97 | \( 1 - 1.74iT - 97T^{2} \) |
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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
−9.742195285219644017756543837177, −8.761172821236083813033257195108, −8.105293957941451299758968513148, −6.99054055601518640029609690712, −6.61744434281939960683331022199, −5.53689104351258673340012056190, −4.60110112443056647207437225889, −4.09528919502682933491981832281, −2.39346153659905976939256762414, −1.66097543125998331286665391643,
0.46146587527738210505991487607, 1.58342406416997017381447604564, 3.10366563592021483723696861665, 4.13122260252501347967302085444, 5.16046429044944981524006554584, 5.57357408908154052711519775392, 6.62213182607485079696155042037, 7.54750513119455769994107430245, 8.344492929929650899812031350831, 8.861081572099015843421654967444
