| L(s) = 1 | + 0.755·2-s + 2.51·3-s − 1.42·4-s + 1.90·6-s − 3.75·7-s − 2.59·8-s + 3.33·9-s − 1.90·11-s − 3.59·12-s − 5.10·13-s − 2.84·14-s + 0.900·16-s − 3.75·17-s + 2.51·18-s − 9.45·21-s − 1.43·22-s + 4.13·23-s − 6.51·24-s − 3.85·26-s + 0.830·27-s + 5.37·28-s + 2.84·29-s + 6.61·31-s + 5.86·32-s − 4.78·33-s − 2.84·34-s − 4.75·36-s + ⋯ |
| L(s) = 1 | + 0.534·2-s + 1.45·3-s − 0.714·4-s + 0.776·6-s − 1.42·7-s − 0.916·8-s + 1.11·9-s − 0.573·11-s − 1.03·12-s − 1.41·13-s − 0.759·14-s + 0.225·16-s − 0.911·17-s + 0.593·18-s − 2.06·21-s − 0.306·22-s + 0.861·23-s − 1.33·24-s − 0.756·26-s + 0.159·27-s + 1.01·28-s + 0.527·29-s + 1.18·31-s + 1.03·32-s − 0.832·33-s − 0.487·34-s − 0.793·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.963896523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.963896523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 0.755T + 2T^{2} \) |
| 3 | \( 1 - 2.51T + 3T^{2} \) |
| 7 | \( 1 + 3.75T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 + 5.10T + 13T^{2} \) |
| 17 | \( 1 + 3.75T + 17T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 - 5.96T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 5.35T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 9.74T + 61T^{2} \) |
| 67 | \( 1 + 5.86T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 0.868T + 73T^{2} \) |
| 79 | \( 1 + 0.655T + 79T^{2} \) |
| 83 | \( 1 - 0.868T + 83T^{2} \) |
| 89 | \( 1 - 3.77T + 89T^{2} \) |
| 97 | \( 1 - 4.09T + 97T^{2} \) |
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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
−7.58813788207839336037840805638, −7.30219352918220323579848789582, −6.32096354786927089454616610214, −5.65954968619355848787057395512, −4.60295127182197183936481454733, −4.28681314297923859728968943949, −3.20003483251365386132070051325, −2.89822395838593746630885947576, −2.28185064688149424346534082038, −0.54402055880211993524530495773,
0.54402055880211993524530495773, 2.28185064688149424346534082038, 2.89822395838593746630885947576, 3.20003483251365386132070051325, 4.28681314297923859728968943949, 4.60295127182197183936481454733, 5.65954968619355848787057395512, 6.32096354786927089454616610214, 7.30219352918220323579848789582, 7.58813788207839336037840805638
