| L(s) = 1 | + 3-s + 1.21·5-s + 5.16·7-s + 9-s + 2.78·11-s − 3.00·13-s + 1.21·15-s − 1.79·17-s − 5.05·19-s + 5.16·21-s + 0.438·23-s − 3.52·25-s + 27-s + 2.79·29-s − 1.21·31-s + 2.78·33-s + 6.27·35-s + 11.4·37-s − 3.00·39-s − 4.44·41-s + 2.22·43-s + 1.21·45-s + 11.1·47-s + 19.6·49-s − 1.79·51-s − 10.2·53-s + 3.38·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.543·5-s + 1.95·7-s + 0.333·9-s + 0.840·11-s − 0.834·13-s + 0.313·15-s − 0.434·17-s − 1.15·19-s + 1.12·21-s + 0.0915·23-s − 0.704·25-s + 0.192·27-s + 0.519·29-s − 0.218·31-s + 0.485·33-s + 1.06·35-s + 1.87·37-s − 0.481·39-s − 0.694·41-s + 0.339·43-s + 0.181·45-s + 1.62·47-s + 2.81·49-s − 0.250·51-s − 1.40·53-s + 0.456·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.720297992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.720297992\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 1.21T + 5T^{2} \) |
| 7 | \( 1 - 5.16T + 7T^{2} \) |
| 11 | \( 1 - 2.78T + 11T^{2} \) |
| 13 | \( 1 + 3.00T + 13T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 - 0.438T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 + 1.21T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 4.44T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 4.40T + 59T^{2} \) |
| 61 | \( 1 + 0.475T + 61T^{2} \) |
| 67 | \( 1 - 0.659T + 67T^{2} \) |
| 71 | \( 1 - 2.62T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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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
−8.014455563780360632232049146106, −7.61848349990350449559284747177, −6.69428091099096410166735395933, −5.96601890757507786081961946979, −4.99781745189756652433632615706, −4.49667426006135253541435533504, −3.81887440978274191303676669643, −2.35049353880183476468736724334, −2.07602516251966712110744919873, −1.05062244315341459260813176276,
1.05062244315341459260813176276, 2.07602516251966712110744919873, 2.35049353880183476468736724334, 3.81887440978274191303676669643, 4.49667426006135253541435533504, 4.99781745189756652433632615706, 5.96601890757507786081961946979, 6.69428091099096410166735395933, 7.61848349990350449559284747177, 8.014455563780360632232049146106
