| L(s) = 1 | − 1.41·2-s − 0.773·3-s + 0.00129·4-s + 1.09·6-s + 2.74·7-s + 2.82·8-s − 2.40·9-s − 1.62·11-s − 0.00100·12-s + 4.03·13-s − 3.88·14-s − 4.00·16-s − 1.28·17-s + 3.39·18-s + 3.97·19-s − 2.12·21-s + 2.30·22-s − 9.42·23-s − 2.18·24-s − 5.70·26-s + 4.18·27-s + 0.00356·28-s + 2.84·29-s − 2.91·31-s + 0.00734·32-s + 1.26·33-s + 1.82·34-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.446·3-s + 0.000649·4-s + 0.446·6-s + 1.03·7-s + 0.999·8-s − 0.800·9-s − 0.491·11-s − 0.000290·12-s + 1.11·13-s − 1.03·14-s − 1.00·16-s − 0.312·17-s + 0.800·18-s + 0.911·19-s − 0.464·21-s + 0.491·22-s − 1.96·23-s − 0.446·24-s − 1.11·26-s + 0.804·27-s + 0.000674·28-s + 0.528·29-s − 0.523·31-s + 0.00129·32-s + 0.219·33-s + 0.312·34-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 + 0.773T + 3T^{2} \) |
| 7 | \( 1 - 2.74T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 13 | \( 1 - 4.03T + 13T^{2} \) |
| 17 | \( 1 + 1.28T + 17T^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 23 | \( 1 + 9.42T + 23T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 - 9.40T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 + 6.52T + 43T^{2} \) |
| 47 | \( 1 + 0.230T + 47T^{2} \) |
| 53 | \( 1 + 6.13T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 8.00T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.30T + 73T^{2} \) |
| 79 | \( 1 + 5.41T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.84482385980194231052852960048, −7.42578672768930478165310816853, −6.11893123612568651799103744629, −5.79615499746334728599787087609, −4.77209418717482714704597714184, −4.27592220022104479076078160558, −3.10230945245189708250848962149, −1.95388802342105134212081696790, −1.12130421719650979560098281044, 0,
1.12130421719650979560098281044, 1.95388802342105134212081696790, 3.10230945245189708250848962149, 4.27592220022104479076078160558, 4.77209418717482714704597714184, 5.79615499746334728599787087609, 6.11893123612568651799103744629, 7.42578672768930478165310816853, 7.84482385980194231052852960048
