| L(s) = 1 | − 2.70·2-s + 0.205·3-s + 5.31·4-s − 0.555·6-s − 2.55·7-s − 8.95·8-s − 2.95·9-s − 11-s + 1.09·12-s − 0.727·13-s + 6.90·14-s + 13.5·16-s + 4.75·17-s + 7.99·18-s + 19-s − 0.523·21-s + 2.70·22-s − 5.27·23-s − 1.83·24-s + 1.96·26-s − 1.22·27-s − 13.5·28-s + 6.31·29-s + 2.60·31-s − 18.8·32-s − 0.205·33-s − 12.8·34-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 0.118·3-s + 2.65·4-s − 0.226·6-s − 0.964·7-s − 3.16·8-s − 0.985·9-s − 0.301·11-s + 0.314·12-s − 0.201·13-s + 1.84·14-s + 3.39·16-s + 1.15·17-s + 1.88·18-s + 0.229·19-s − 0.114·21-s + 0.576·22-s − 1.09·23-s − 0.375·24-s + 0.385·26-s − 0.235·27-s − 2.56·28-s + 1.17·29-s + 0.467·31-s − 3.33·32-s − 0.0357·33-s − 2.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 - 0.205T + 3T^{2} \) |
| 7 | \( 1 + 2.55T + 7T^{2} \) |
| 13 | \( 1 + 0.727T + 13T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 23 | \( 1 + 5.27T + 23T^{2} \) |
| 29 | \( 1 - 6.31T + 29T^{2} \) |
| 31 | \( 1 - 2.60T + 31T^{2} \) |
| 37 | \( 1 + 7.54T + 37T^{2} \) |
| 41 | \( 1 - 6.59T + 41T^{2} \) |
| 43 | \( 1 + 0.578T + 43T^{2} \) |
| 47 | \( 1 - 5.00T + 47T^{2} \) |
| 53 | \( 1 + 5.82T + 53T^{2} \) |
| 59 | \( 1 - 8.71T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 - 9.50T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 8.52T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.070398661114127615377608519652, −7.39843528058669559711124318007, −6.60694443233227980901564196922, −6.03921030130057258363974234409, −5.30911303895964068280225525460, −3.64365764133469213862036991898, −2.89873325125029541981637100318, −2.26222089836759792475795270645, −0.980453479666625752880055920239, 0,
0.980453479666625752880055920239, 2.26222089836759792475795270645, 2.89873325125029541981637100318, 3.64365764133469213862036991898, 5.30911303895964068280225525460, 6.03921030130057258363974234409, 6.60694443233227980901564196922, 7.39843528058669559711124318007, 8.070398661114127615377608519652
