| L(s) = 1 | − 2.41·2-s − 1.41·3-s + 3.82·4-s + 5-s + 3.41·6-s + 4.82·7-s − 4.41·8-s − 0.999·9-s − 2.41·10-s + 3.41·11-s − 5.41·12-s − 13-s − 11.6·14-s − 1.41·15-s + 2.99·16-s + 0.828·17-s + 2.41·18-s + 0.585·19-s + 3.82·20-s − 6.82·21-s − 8.24·22-s + 1.41·23-s + 6.24·24-s + 25-s + 2.41·26-s + 5.65·27-s + 18.4·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.816·3-s + 1.91·4-s + 0.447·5-s + 1.39·6-s + 1.82·7-s − 1.56·8-s − 0.333·9-s − 0.763·10-s + 1.02·11-s − 1.56·12-s − 0.277·13-s − 3.11·14-s − 0.365·15-s + 0.749·16-s + 0.200·17-s + 0.569·18-s + 0.134·19-s + 0.856·20-s − 1.49·21-s − 1.75·22-s + 0.294·23-s + 1.27·24-s + 0.200·25-s + 0.473·26-s + 1.08·27-s + 3.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4252471420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4252471420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 - 2.48T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 3.17T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−15.12084211380259290331308357280, −14.09788229798620988385387724562, −11.87625824790741562856339087337, −11.33974627039112642463347181324, −10.38769790595884851814500436599, −9.048857849602209042479928999365, −8.096743379908091249165589220980, −6.76239892610010783175993067675, −5.19418508234729815424437434398, −1.60142152871881162770674194405,
1.60142152871881162770674194405, 5.19418508234729815424437434398, 6.76239892610010783175993067675, 8.096743379908091249165589220980, 9.048857849602209042479928999365, 10.38769790595884851814500436599, 11.33974627039112642463347181324, 11.87625824790741562856339087337, 14.09788229798620988385387724562, 15.12084211380259290331308357280
