| L(s) = 1 | + 3-s + 3.82·5-s − 3.82·7-s − 2·9-s − 4.82·11-s − 13-s + 3.82·15-s + 6.65·17-s + 4·19-s − 3.82·21-s + 3.17·23-s + 9.65·25-s − 5·27-s + 3.17·29-s − 4.82·33-s − 14.6·35-s + 3.82·37-s − 39-s + 2.82·41-s + 3·43-s − 7.65·45-s + 11.4·47-s + 7.65·49-s + 6.65·51-s + 3.17·53-s − 18.4·55-s + 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.71·5-s − 1.44·7-s − 0.666·9-s − 1.45·11-s − 0.277·13-s + 0.988·15-s + 1.61·17-s + 0.917·19-s − 0.835·21-s + 0.661·23-s + 1.93·25-s − 0.962·27-s + 0.588·29-s − 0.840·33-s − 2.47·35-s + 0.629·37-s − 0.160·39-s + 0.441·41-s + 0.457·43-s − 1.14·45-s + 1.67·47-s + 1.09·49-s + 0.932·51-s + 0.435·53-s − 2.49·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.467615150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.467615150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 17 | \( 1 - 6.65T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 - 2.82T + 41T^{2} \) |
| 43 | \( 1 - 3T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.65T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.836160246933541023025524861342, −7.88148280366647370037789780362, −7.15982444368058360942511401733, −6.21308107924868045330439706988, −5.53878001419693789328750441911, −5.25660869875266511020211482117, −3.57980207144784987208560444811, −2.66269750149684415213095649642, −2.57265194183661925342023720872, −0.902068757872075446271903384236,
0.902068757872075446271903384236, 2.57265194183661925342023720872, 2.66269750149684415213095649642, 3.57980207144784987208560444811, 5.25660869875266511020211482117, 5.53878001419693789328750441911, 6.21308107924868045330439706988, 7.15982444368058360942511401733, 7.88148280366647370037789780362, 8.836160246933541023025524861342
