L(s) = 1 | + 2.41·2-s + 3-s + 3.82·4-s + 5-s + 2.41·6-s + 4.41·8-s + 9-s + 2.41·10-s + 2.82·11-s + 3.82·12-s − 4.82·13-s + 15-s + 2.99·16-s − 7.65·17-s + 2.41·18-s − 0.828·19-s + 3.82·20-s + 6.82·22-s − 7.65·23-s + 4.41·24-s + 25-s − 11.6·26-s + 27-s + 6·29-s + 2.41·30-s + 6.48·31-s − 1.58·32-s + ⋯ |
L(s) = 1 | + 1.70·2-s + 0.577·3-s + 1.91·4-s + 0.447·5-s + 0.985·6-s + 1.56·8-s + 0.333·9-s + 0.763·10-s + 0.852·11-s + 1.10·12-s − 1.33·13-s + 0.258·15-s + 0.749·16-s − 1.85·17-s + 0.569·18-s − 0.190·19-s + 0.856·20-s + 1.45·22-s − 1.59·23-s + 0.901·24-s + 0.200·25-s − 2.28·26-s + 0.192·27-s + 1.11·29-s + 0.440·30-s + 1.16·31-s − 0.280·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.815584492\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.815584492\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 - 6.48T + 97T^{2} \) |
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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
−10.54625157625561970359324362460, −9.588522713859397681651882030334, −8.684998343074769592528877385470, −7.41901133486708699633132511733, −6.57911247197191112179156152209, −5.90802503008984214810914251076, −4.53096583852433945168714755738, −4.25069545217165665455890780164, −2.77490002575291295927555861610, −2.09963858328191631751420322491,
2.09963858328191631751420322491, 2.77490002575291295927555861610, 4.25069545217165665455890780164, 4.53096583852433945168714755738, 5.90802503008984214810914251076, 6.57911247197191112179156152209, 7.41901133486708699633132511733, 8.684998343074769592528877385470, 9.588522713859397681651882030334, 10.54625157625561970359324362460