| L(s) = 1 | + 2.19·2-s + 1.13·3-s + 2.80·4-s − 3.87·5-s + 2.49·6-s − 3.34·7-s + 1.77·8-s − 1.70·9-s − 8.50·10-s + 3.19·12-s − 3.33·13-s − 7.34·14-s − 4.40·15-s − 1.72·16-s − 2.42·17-s − 3.74·18-s + 7.48·19-s − 10.8·20-s − 3.80·21-s + 6.20·23-s + 2.01·24-s + 10.0·25-s − 7.30·26-s − 5.35·27-s − 9.40·28-s + 2.67·29-s − 9.65·30-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 0.655·3-s + 1.40·4-s − 1.73·5-s + 1.01·6-s − 1.26·7-s + 0.627·8-s − 0.569·9-s − 2.68·10-s + 0.921·12-s − 0.923·13-s − 1.96·14-s − 1.13·15-s − 0.431·16-s − 0.588·17-s − 0.883·18-s + 1.71·19-s − 2.43·20-s − 0.829·21-s + 1.29·23-s + 0.411·24-s + 2.00·25-s − 1.43·26-s − 1.02·27-s − 1.77·28-s + 0.496·29-s − 1.76·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.663360293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.663360293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 3 | \( 1 - 1.13T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 13 | \( 1 + 3.33T + 13T^{2} \) |
| 17 | \( 1 + 2.42T + 17T^{2} \) |
| 19 | \( 1 - 7.48T + 19T^{2} \) |
| 23 | \( 1 - 6.20T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 - 8.31T + 31T^{2} \) |
| 37 | \( 1 - 5.59T + 37T^{2} \) |
| 41 | \( 1 + 0.632T + 41T^{2} \) |
| 43 | \( 1 + 5.14T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 0.797T + 59T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 6.06T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 3.95T + 97T^{2} \) |
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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
−7.77202754685938119862388021162, −6.95608793333403279670490166428, −6.66049160484127722907918733700, −5.59696269690640863030100526954, −4.85812090321731059105255081511, −4.26560433904396593588120073015, −3.41165958899579174484144699627, −3.05959139304582640393841289957, −2.61565654141577367359197524553, −0.58925841748959591650332534198,
0.58925841748959591650332534198, 2.61565654141577367359197524553, 3.05959139304582640393841289957, 3.41165958899579174484144699627, 4.26560433904396593588120073015, 4.85812090321731059105255081511, 5.59696269690640863030100526954, 6.66049160484127722907918733700, 6.95608793333403279670490166428, 7.77202754685938119862388021162
