| L(s) = 1 | − 2.04·2-s + 3.19·3-s + 2.18·4-s − 6.53·6-s − 1.17·7-s − 0.372·8-s + 7.21·9-s − 4.92·11-s + 6.97·12-s + 2.46·13-s + 2.40·14-s − 3.60·16-s − 14.7·18-s + 2.04·19-s − 3.76·21-s + 10.0·22-s + 0.119·23-s − 1.19·24-s − 5.03·26-s + 13.4·27-s − 2.56·28-s − 1.06·29-s − 2.79·31-s + 8.11·32-s − 15.7·33-s + 15.7·36-s − 2.31·37-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 1.84·3-s + 1.09·4-s − 2.66·6-s − 0.445·7-s − 0.131·8-s + 2.40·9-s − 1.48·11-s + 2.01·12-s + 0.683·13-s + 0.643·14-s − 0.900·16-s − 3.47·18-s + 0.468·19-s − 0.821·21-s + 2.14·22-s + 0.0248·23-s − 0.243·24-s − 0.987·26-s + 2.59·27-s − 0.485·28-s − 0.197·29-s − 0.502·31-s + 1.43·32-s − 2.73·33-s + 2.62·36-s − 0.380·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.765003003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.765003003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.04T + 2T^{2} \) |
| 3 | \( 1 - 3.19T + 3T^{2} \) |
| 7 | \( 1 + 1.17T + 7T^{2} \) |
| 11 | \( 1 + 4.92T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 - 0.119T + 23T^{2} \) |
| 29 | \( 1 + 1.06T + 29T^{2} \) |
| 31 | \( 1 + 2.79T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + 0.717T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 3.39T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 - 1.51T + 59T^{2} \) |
| 61 | \( 1 + 8.08T + 61T^{2} \) |
| 67 | \( 1 - 4.92T + 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 - 8.46T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
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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
−8.087667603823580762573100494299, −7.44656938742285807593300027288, −7.22338796206369774758957832728, −6.10469623074997039254150174750, −5.00361191108943343864784818647, −4.01762224505859485507203590832, −3.23891471909947229442061417324, −2.50264420937711027165670946581, −1.87367044370317025595944866379, −0.76181086935062550358849177000,
0.76181086935062550358849177000, 1.87367044370317025595944866379, 2.50264420937711027165670946581, 3.23891471909947229442061417324, 4.01762224505859485507203590832, 5.00361191108943343864784818647, 6.10469623074997039254150174750, 7.22338796206369774758957832728, 7.44656938742285807593300027288, 8.087667603823580762573100494299
