L(s) = 1 | + 2.67·2-s − 1.67·3-s + 5.15·4-s + 2.86·5-s − 4.46·6-s − 5.21·7-s + 8.44·8-s − 0.209·9-s + 7.66·10-s + 0.107·11-s − 8.61·12-s − 13.9·14-s − 4.78·15-s + 12.2·16-s + 1.65·17-s − 0.560·18-s + 2.74·19-s + 14.7·20-s + 8.70·21-s + 0.287·22-s − 2.05·23-s − 14.1·24-s + 3.21·25-s + 5.36·27-s − 26.8·28-s + 8.54·29-s − 12.8·30-s + ⋯ |
L(s) = 1 | + 1.89·2-s − 0.964·3-s + 2.57·4-s + 1.28·5-s − 1.82·6-s − 1.96·7-s + 2.98·8-s − 0.0698·9-s + 2.42·10-s + 0.0323·11-s − 2.48·12-s − 3.72·14-s − 1.23·15-s + 3.06·16-s + 0.401·17-s − 0.132·18-s + 0.630·19-s + 3.30·20-s + 1.89·21-s + 0.0612·22-s − 0.427·23-s − 2.87·24-s + 0.642·25-s + 1.03·27-s − 5.07·28-s + 1.58·29-s − 2.33·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.223495174\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.223495174\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 - 2.86T + 5T^{2} \) |
| 7 | \( 1 + 5.21T + 7T^{2} \) |
| 11 | \( 1 - 0.107T + 11T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 - 8.54T + 29T^{2} \) |
| 37 | \( 1 - 7.15T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 43 | \( 1 - 7.32T + 43T^{2} \) |
| 47 | \( 1 - 4.70T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 - 8.50T + 71T^{2} \) |
| 73 | \( 1 - 6.98T + 73T^{2} \) |
| 79 | \( 1 + 1.90T + 79T^{2} \) |
| 83 | \( 1 - 9.64T + 83T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.74699460806194879233946726565, −6.82838119170654600709540792504, −6.26831309098229731265635020092, −5.99607177027027930492252355817, −5.53164689828565401741453071696, −4.68844247262488856166047989809, −3.78381926246443651040759736802, −2.86532026903669651321483104450, −2.50041817609849460668581570945, −0.981314568525169403896780367462,
0.981314568525169403896780367462, 2.50041817609849460668581570945, 2.86532026903669651321483104450, 3.78381926246443651040759736802, 4.68844247262488856166047989809, 5.53164689828565401741453071696, 5.99607177027027930492252355817, 6.26831309098229731265635020092, 6.82838119170654600709540792504, 7.74699460806194879233946726565