L(s) = 1 | + (−1.66 + 1.66i)3-s + (−2.27 + 1.34i)7-s − 2.56i·9-s + 1.56·11-s + (−2.60 + 2.60i)13-s + (−2.60 − 2.60i)17-s − 2.64·19-s + (1.56 − 6.04i)21-s + (0.794 + 0.794i)23-s + (−0.731 − 0.731i)27-s − 6.68i·29-s + 9.43i·31-s + (−2.60 + 2.60i)33-s + (2.82 − 2.82i)37-s − 8.68i·39-s + ⋯ |
L(s) = 1 | + (−0.962 + 0.962i)3-s + (−0.861 + 0.507i)7-s − 0.853i·9-s + 0.470·11-s + (−0.722 + 0.722i)13-s + (−0.631 − 0.631i)17-s − 0.607·19-s + (0.340 − 1.31i)21-s + (0.165 + 0.165i)23-s + (−0.140 − 0.140i)27-s − 1.24i·29-s + 1.69i·31-s + (−0.453 + 0.453i)33-s + (0.464 − 0.464i)37-s − 1.39i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2382468883\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2382468883\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.27 - 1.34i)T \) |
good | 3 | \( 1 + (1.66 - 1.66i)T - 3iT^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + (2.60 - 2.60i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.60 + 2.60i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + (-0.794 - 0.794i)T + 23iT^{2} \) |
| 29 | \( 1 + 6.68iT - 29T^{2} \) |
| 31 | \( 1 - 9.43iT - 31T^{2} \) |
| 37 | \( 1 + (-2.82 + 2.82i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.64iT - 41T^{2} \) |
| 43 | \( 1 + (-6.45 - 6.45i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.66 + 1.66i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.24 + 7.24i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 9.43iT - 61T^{2} \) |
| 67 | \( 1 + (3.62 - 3.62i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + (-6.67 + 6.67i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.8iT - 79T^{2} \) |
| 83 | \( 1 + (-9.47 + 9.47i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.43T + 89T^{2} \) |
| 97 | \( 1 + (-5.52 - 5.52i)T + 97iT^{2} \) |
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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
−9.345850385179208688572938274533, −9.105030935774286188656087207675, −7.72413467779686727249544009704, −6.56646893501266572504675473223, −6.20067939150176391073869028885, −5.05073676018519207649443357140, −4.50822431692988102932628593527, −3.46235090899308118435988991821, −2.21574912088882408240270482758, −0.12644937876407215940539991924,
1.05509267414508224379523946744, 2.44277665838068231919145665501, 3.72814019895544339562061011833, 4.77915043798701855666495726401, 5.96090151064863516469699195247, 6.35990869686883016045524974289, 7.18172835001154207928206684912, 7.80827506869126108648294656590, 8.976516450823781224420779522199, 9.782367912572601214674145329692