| L(s) = 1 | + 2.70·3-s − 5-s + 3.76·7-s + 4.32·9-s + 3.26·13-s − 2.70·15-s − 3.90·17-s + 4.32·19-s + 10.1·21-s − 6.41·23-s + 25-s + 3.58·27-s + 6.50·29-s + 5.48·31-s − 3.76·35-s − 10.0·37-s + 8.84·39-s + 8.01·41-s + 0.906·43-s − 4.32·45-s + 5.52·47-s + 7.14·49-s − 10.5·51-s + 1.09·53-s + 11.7·57-s − 6.63·59-s − 9.02·61-s + ⋯ |
| L(s) = 1 | + 1.56·3-s − 0.447·5-s + 1.42·7-s + 1.44·9-s + 0.906·13-s − 0.698·15-s − 0.947·17-s + 0.992·19-s + 2.22·21-s − 1.33·23-s + 0.200·25-s + 0.689·27-s + 1.20·29-s + 0.985·31-s − 0.635·35-s − 1.64·37-s + 1.41·39-s + 1.25·41-s + 0.138·43-s − 0.644·45-s + 0.806·47-s + 1.02·49-s − 1.48·51-s + 0.149·53-s + 1.55·57-s − 0.863·59-s − 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.617585936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.617585936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.70T + 3T^{2} \) |
| 7 | \( 1 - 3.76T + 7T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 + 6.41T + 23T^{2} \) |
| 29 | \( 1 - 6.50T + 29T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 - 0.906T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 - 1.09T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 + 9.02T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 - 7.27T + 79T^{2} \) |
| 83 | \( 1 + 6.11T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.637896112217821063940818014859, −8.307505484158548854525965028894, −7.73553471351636110445916729279, −6.96371307932995727196743755986, −5.81654844609874087404805973023, −4.63010398893804890204920370373, −4.10053036302398818579616646595, −3.15066727866729169187215237832, −2.21293603170613638497410576225, −1.29044867656301575574357442603,
1.29044867656301575574357442603, 2.21293603170613638497410576225, 3.15066727866729169187215237832, 4.10053036302398818579616646595, 4.63010398893804890204920370373, 5.81654844609874087404805973023, 6.96371307932995727196743755986, 7.73553471351636110445916729279, 8.307505484158548854525965028894, 8.637896112217821063940818014859
