| L(s) = 1 | − 1.44·3-s + 2.86·7-s − 0.909·9-s + 3.84·11-s − 6.22·13-s + 17-s − 6.62·19-s − 4.14·21-s + 4.51·23-s + 5.65·27-s − 0.658·29-s − 3.49·31-s − 5.56·33-s − 3.34·37-s + 8.99·39-s + 2.04·41-s − 1.29·43-s + 6.22·47-s + 1.21·49-s − 1.44·51-s + 9.92·53-s + 9.57·57-s + 2·59-s − 7.61·61-s − 2.60·63-s + 0.257·67-s − 6.53·69-s + ⋯ |
| L(s) = 1 | − 0.834·3-s + 1.08·7-s − 0.303·9-s + 1.15·11-s − 1.72·13-s + 0.242·17-s − 1.51·19-s − 0.904·21-s + 0.942·23-s + 1.08·27-s − 0.122·29-s − 0.628·31-s − 0.968·33-s − 0.549·37-s + 1.44·39-s + 0.318·41-s − 0.197·43-s + 0.907·47-s + 0.172·49-s − 0.202·51-s + 1.36·53-s + 1.26·57-s + 0.260·59-s − 0.975·61-s − 0.328·63-s + 0.0314·67-s − 0.786·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.299695663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.299695663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.44T + 3T^{2} \) |
| 7 | \( 1 - 2.86T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 19 | \( 1 + 6.62T + 19T^{2} \) |
| 23 | \( 1 - 4.51T + 23T^{2} \) |
| 29 | \( 1 + 0.658T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 + 3.34T + 37T^{2} \) |
| 41 | \( 1 - 2.04T + 41T^{2} \) |
| 43 | \( 1 + 1.29T + 43T^{2} \) |
| 47 | \( 1 - 6.22T + 47T^{2} \) |
| 53 | \( 1 - 9.92T + 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + 7.61T + 61T^{2} \) |
| 67 | \( 1 - 0.257T + 67T^{2} \) |
| 71 | \( 1 + 1.18T + 71T^{2} \) |
| 73 | \( 1 + 3.26T + 73T^{2} \) |
| 79 | \( 1 - 4.99T + 79T^{2} \) |
| 83 | \( 1 - 7.91T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−7.85322014100423969788381821661, −7.21096895390957029815928567875, −6.55935904091557270384602045763, −5.82291069246123159861688864695, −5.01363483894396269443476257077, −4.66691456551665566088945258548, −3.75960132275036766151816934272, −2.56238257598738840075153297259, −1.77105534131034895201135926071, −0.60873149548084074264631979494,
0.60873149548084074264631979494, 1.77105534131034895201135926071, 2.56238257598738840075153297259, 3.75960132275036766151816934272, 4.66691456551665566088945258548, 5.01363483894396269443476257077, 5.82291069246123159861688864695, 6.55935904091557270384602045763, 7.21096895390957029815928567875, 7.85322014100423969788381821661
