| L(s) = 1 | − 2.35i·2-s − 3.53·4-s + i·5-s + 3.60i·7-s + 3.61i·8-s + 2.35·10-s − 4.88i·11-s + (2.43 + 2.66i)13-s + 8.47·14-s + 1.43·16-s − 6.75·17-s − 5.81i·19-s − 3.53i·20-s − 11.5·22-s − 2.87·23-s + ⋯ |
| L(s) = 1 | − 1.66i·2-s − 1.76·4-s + 0.447i·5-s + 1.36i·7-s + 1.27i·8-s + 0.744·10-s − 1.47i·11-s + (0.674 + 0.738i)13-s + 2.26·14-s + 0.357·16-s − 1.63·17-s − 1.33i·19-s − 0.790i·20-s − 2.45·22-s − 0.598·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5083885410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5083885410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-2.43 - 2.66i)T \) |
| good | 2 | \( 1 + 2.35iT - 2T^{2} \) |
| 7 | \( 1 - 3.60iT - 7T^{2} \) |
| 11 | \( 1 + 4.88iT - 11T^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 19 | \( 1 + 5.81iT - 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 - 0.659T + 29T^{2} \) |
| 31 | \( 1 - 9.20iT - 31T^{2} \) |
| 37 | \( 1 - 8.19iT - 37T^{2} \) |
| 41 | \( 1 + 3.18iT - 41T^{2} \) |
| 43 | \( 1 + 5.44T + 43T^{2} \) |
| 47 | \( 1 - 6.79iT - 47T^{2} \) |
| 53 | \( 1 - 2.38T + 53T^{2} \) |
| 59 | \( 1 - 8.50iT - 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 + 1.85iT - 71T^{2} \) |
| 73 | \( 1 + 8.47iT - 73T^{2} \) |
| 79 | \( 1 + 8.71T + 79T^{2} \) |
| 83 | \( 1 - 3.38iT - 83T^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 + 11.2iT - 97T^{2} \) |
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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
−9.350117706802483893718868206821, −8.734347159691519739307183094803, −8.554923261826429973292488081073, −6.79860911511187376521453535416, −6.14775904828632591712657939918, −5.04864656272027033373375661382, −4.13846651897136217768717056233, −3.03929549984422345016204266102, −2.59292027489926682346480576085, −1.47341053487433590677091970294,
0.18885272474864745184448975972, 1.89076090986937723222988975241, 3.97673978909626124685469695816, 4.29948790648030398409709026576, 5.26812596716759203756084309865, 6.20226453875947257862717168701, 6.83999013884511628128041631298, 7.68444019344812069477308746021, 7.999763086675098780429242019014, 8.979217047169985395401980145582
