| L(s) = 1 | − 1.54·2-s − 2.81·3-s + 0.398·4-s + 4.35·6-s + 4.24·7-s + 2.47·8-s + 4.90·9-s + 0.915·11-s − 1.12·12-s − 4.81·13-s − 6.57·14-s − 4.63·16-s + 5.38·17-s − 7.59·18-s − 4.34·19-s − 11.9·21-s − 1.41·22-s − 8.10·23-s − 6.97·24-s + 7.45·26-s − 5.34·27-s + 1.69·28-s − 6.45·29-s + 10.7·31-s + 2.22·32-s − 2.57·33-s − 8.34·34-s + ⋯ |
| L(s) = 1 | − 1.09·2-s − 1.62·3-s + 0.199·4-s + 1.77·6-s + 1.60·7-s + 0.876·8-s + 1.63·9-s + 0.276·11-s − 0.323·12-s − 1.33·13-s − 1.75·14-s − 1.15·16-s + 1.30·17-s − 1.78·18-s − 0.997·19-s − 2.60·21-s − 0.302·22-s − 1.69·23-s − 1.42·24-s + 1.46·26-s − 1.02·27-s + 0.319·28-s − 1.19·29-s + 1.93·31-s + 0.393·32-s − 0.447·33-s − 1.43·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4905642059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4905642059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.54T + 2T^{2} \) |
| 3 | \( 1 + 2.81T + 3T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 0.915T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 - 5.38T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 + 8.10T + 23T^{2} \) |
| 29 | \( 1 + 6.45T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 + 5.16T + 37T^{2} \) |
| 41 | \( 1 + 0.612T + 41T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 - 2.21T + 47T^{2} \) |
| 53 | \( 1 + 0.00846T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 - 1.48T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 + 7.45T + 83T^{2} \) |
| 89 | \( 1 + 0.520T + 89T^{2} \) |
| 97 | \( 1 - 6.33T + 97T^{2} \) |
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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
−8.040468046723424358027273412779, −7.50291384883029573159778799725, −6.83478544098632697730527105972, −5.84741551776883485929205840365, −5.26488126375784386549407764930, −4.59934491311818195272861161603, −4.09333825016662926515373073917, −2.18857653605444168935949099546, −1.45869755532248128003965703726, −0.51385270651360754836525678170,
0.51385270651360754836525678170, 1.45869755532248128003965703726, 2.18857653605444168935949099546, 4.09333825016662926515373073917, 4.59934491311818195272861161603, 5.26488126375784386549407764930, 5.84741551776883485929205840365, 6.83478544098632697730527105972, 7.50291384883029573159778799725, 8.040468046723424358027273412779
