L(s) = 1 | − 1.08·2-s + 3-s − 0.830·4-s − 1.66·5-s − 1.08·6-s − 7-s + 3.06·8-s + 9-s + 1.80·10-s + 4.63·11-s − 0.830·12-s + 1.08·14-s − 1.66·15-s − 1.64·16-s + 4.03·17-s − 1.08·18-s + 0.793·19-s + 1.38·20-s − 21-s − 5.01·22-s + 4.31·23-s + 3.06·24-s − 2.21·25-s + 27-s + 0.830·28-s + 4.61·29-s + 1.80·30-s + ⋯ |
L(s) = 1 | − 0.764·2-s + 0.577·3-s − 0.415·4-s − 0.746·5-s − 0.441·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s + 0.570·10-s + 1.39·11-s − 0.239·12-s + 0.288·14-s − 0.430·15-s − 0.411·16-s + 0.977·17-s − 0.254·18-s + 0.182·19-s + 0.310·20-s − 0.218·21-s − 1.06·22-s + 0.899·23-s + 0.624·24-s − 0.442·25-s + 0.192·27-s + 0.157·28-s + 0.857·29-s + 0.329·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.201686757\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.201686757\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.08T + 2T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 17 | \( 1 - 4.03T + 17T^{2} \) |
| 19 | \( 1 - 0.793T + 19T^{2} \) |
| 23 | \( 1 - 4.31T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 + 5.75T + 31T^{2} \) |
| 37 | \( 1 + 5.73T + 37T^{2} \) |
| 41 | \( 1 + 7.33T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 5.10T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 1.86T + 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 - 3.11T + 73T^{2} \) |
| 79 | \( 1 - 2.28T + 79T^{2} \) |
| 83 | \( 1 + 0.655T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.705840044240023545250390625473, −7.912467067480556210729300376728, −7.34419981263906428674902800100, −6.66673745742666240313818091326, −5.53058080250058724906777399340, −4.52219498634741231669669679079, −3.80311146636830793196719386166, −3.20657567835100196539480935947, −1.70318741793399890171852194183, −0.74995212947272138731747924491,
0.74995212947272138731747924491, 1.70318741793399890171852194183, 3.20657567835100196539480935947, 3.80311146636830793196719386166, 4.52219498634741231669669679079, 5.53058080250058724906777399340, 6.66673745742666240313818091326, 7.34419981263906428674902800100, 7.912467067480556210729300376728, 8.705840044240023545250390625473