L(s) = 1 | − 2-s + 0.129·3-s + 4-s − 3.52·5-s − 0.129·6-s − 2.82·7-s − 8-s − 2.98·9-s + 3.52·10-s + 3.84·11-s + 0.129·12-s − 4.00·13-s + 2.82·14-s − 0.455·15-s + 16-s + 0.713·17-s + 2.98·18-s + 19-s − 3.52·20-s − 0.364·21-s − 3.84·22-s − 5.48·23-s − 0.129·24-s + 7.43·25-s + 4.00·26-s − 0.772·27-s − 2.82·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.0745·3-s + 0.5·4-s − 1.57·5-s − 0.0527·6-s − 1.06·7-s − 0.353·8-s − 0.994·9-s + 1.11·10-s + 1.15·11-s + 0.0372·12-s − 1.11·13-s + 0.755·14-s − 0.117·15-s + 0.250·16-s + 0.173·17-s + 0.703·18-s + 0.229·19-s − 0.788·20-s − 0.0796·21-s − 0.820·22-s − 1.14·23-s − 0.0263·24-s + 1.48·25-s + 0.786·26-s − 0.148·27-s − 0.534·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 0.129T + 3T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 4.00T + 13T^{2} \) |
| 17 | \( 1 - 0.713T + 17T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 - 2.59T + 29T^{2} \) |
| 31 | \( 1 + 1.63T + 31T^{2} \) |
| 37 | \( 1 - 3.57T + 37T^{2} \) |
| 41 | \( 1 - 9.15T + 41T^{2} \) |
| 43 | \( 1 - 7.97T + 43T^{2} \) |
| 47 | \( 1 - 0.365T + 47T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 - 2.96T + 59T^{2} \) |
| 61 | \( 1 - 4.10T + 61T^{2} \) |
| 67 | \( 1 - 8.18T + 67T^{2} \) |
| 71 | \( 1 - 4.50T + 71T^{2} \) |
| 73 | \( 1 - 1.05T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 - 1.95T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 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
−7.60728578082446907355904907205, −6.96076843435517261073261463957, −6.31388207930651457481348832081, −5.58636843024726100631039356256, −4.40628730433231754216145504314, −3.80271748816626981636866477765, −3.09204579957242237458274977966, −2.36045755294888018325948202475, −0.813193970738650805833034310793, 0,
0.813193970738650805833034310793, 2.36045755294888018325948202475, 3.09204579957242237458274977966, 3.80271748816626981636866477765, 4.40628730433231754216145504314, 5.58636843024726100631039356256, 6.31388207930651457481348832081, 6.96076843435517261073261463957, 7.60728578082446907355904907205