L(s) = 1 | + (−1.53 + 0.795i)3-s + 3.80·5-s + (1.73 − 2.44i)9-s + 0.357i·11-s + 4.04i·13-s + (−5.84 + 3.02i)15-s + 0.103·17-s − 2.45i·19-s + 1.33i·23-s + 9.44·25-s + (−0.723 + 5.14i)27-s + 4.97i·29-s + 7.88i·31-s + (−0.284 − 0.549i)33-s − 10.9·37-s + ⋯ |
L(s) = 1 | + (−0.888 + 0.459i)3-s + 1.69·5-s + (0.578 − 0.815i)9-s + 0.107i·11-s + 1.12i·13-s + (−1.50 + 0.780i)15-s + 0.0252·17-s − 0.563i·19-s + 0.277i·23-s + 1.88·25-s + (−0.139 + 0.990i)27-s + 0.923i·29-s + 1.41i·31-s + (−0.0494 − 0.0957i)33-s − 1.79·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.784025778\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.784025778\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.53 - 0.795i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.80T + 5T^{2} \) |
| 11 | \( 1 - 0.357iT - 11T^{2} \) |
| 13 | \( 1 - 4.04iT - 13T^{2} \) |
| 17 | \( 1 - 0.103T + 17T^{2} \) |
| 19 | \( 1 + 2.45iT - 19T^{2} \) |
| 23 | \( 1 - 1.33iT - 23T^{2} \) |
| 29 | \( 1 - 4.97iT - 29T^{2} \) |
| 31 | \( 1 - 7.88iT - 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 + 0.502T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 5.86iT - 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 61 | \( 1 + 9.47iT - 61T^{2} \) |
| 67 | \( 1 + 2.68T + 67T^{2} \) |
| 71 | \( 1 + 5.78iT - 71T^{2} \) |
| 73 | \( 1 + 0.235iT - 73T^{2} \) |
| 79 | \( 1 + 3.22T + 79T^{2} \) |
| 83 | \( 1 - 9.07T + 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 + 5.14iT - 97T^{2} \) |
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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
−9.087777016090649558877852377398, −8.917332199927033556287688269484, −7.17188362153064838435443351069, −6.70132631466172629628246463479, −5.93675886603421739369862356490, −5.23679616994836697046294586199, −4.63567051089703567964398341326, −3.44039350287403411356879735345, −2.15953960976355953842201159308, −1.24717333038836533399089220726,
0.73609134197326094903676878877, 1.88115341868601045010403604718, 2.66038412285582785333778277143, 4.13748531653234237809048912114, 5.35244742716204912801934888164, 5.67137916204050874406231829232, 6.27767793608148530916298562143, 7.14529781538152643218744087826, 7.982057784339710923936641093141, 8.902396667767304028928323900702