| L(s) = 1 | − 1.70·3-s − 5-s − 7-s − 0.0783·9-s + 11-s − 3.70·13-s + 1.70·15-s + 3.70·17-s + 5.26·19-s + 1.70·21-s + 2.34·23-s + 25-s + 5.26·27-s − 5.41·29-s − 8.04·31-s − 1.70·33-s + 35-s − 3.07·37-s + 6.34·39-s − 6.04·41-s + 8.49·43-s + 0.0783·45-s − 10.3·47-s + 49-s − 6.34·51-s − 7.75·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.986·3-s − 0.447·5-s − 0.377·7-s − 0.0261·9-s + 0.301·11-s − 1.02·13-s + 0.441·15-s + 0.899·17-s + 1.20·19-s + 0.372·21-s + 0.487·23-s + 0.200·25-s + 1.01·27-s − 1.00·29-s − 1.44·31-s − 0.297·33-s + 0.169·35-s − 0.506·37-s + 1.01·39-s − 0.944·41-s + 1.29·43-s + 0.0116·45-s − 1.51·47-s + 0.142·49-s − 0.887·51-s − 1.06·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7048601718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7048601718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 13 | \( 1 + 3.70T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 8.04T + 31T^{2} \) |
| 37 | \( 1 + 3.07T + 37T^{2} \) |
| 41 | \( 1 + 6.04T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 7.75T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 + 7.89T + 61T^{2} \) |
| 67 | \( 1 - 3.81T + 67T^{2} \) |
| 71 | \( 1 - 3.41T + 71T^{2} \) |
| 73 | \( 1 + 1.86T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 2.73T + 83T^{2} \) |
| 89 | \( 1 - 0.523T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−7.81539510973211824241690607296, −7.30410569688686018213231047770, −6.69317183938100383257470022445, −5.74624640568119092644954317986, −5.32628409059000295068198402656, −4.63776434889727629650921541783, −3.53956769579326559763625599093, −2.99544319412110645706439645212, −1.64107551509403398252567638938, −0.46391255590274882827430540024,
0.46391255590274882827430540024, 1.64107551509403398252567638938, 2.99544319412110645706439645212, 3.53956769579326559763625599093, 4.63776434889727629650921541783, 5.32628409059000295068198402656, 5.74624640568119092644954317986, 6.69317183938100383257470022445, 7.30410569688686018213231047770, 7.81539510973211824241690607296
