| L(s) = 1 | − 2i·2-s − 4·4-s − 23i·7-s + 8i·8-s − 30·11-s − 34i·13-s − 46·14-s + 16·16-s − 42i·17-s + 139·19-s + 60i·22-s − 192i·23-s − 68·26-s + 92i·28-s + 234·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 1.24i·7-s + 0.353i·8-s − 0.822·11-s − 0.725i·13-s − 0.878·14-s + 0.250·16-s − 0.599i·17-s + 1.67·19-s + 0.581i·22-s − 1.74i·23-s − 0.512·26-s + 0.620i·28-s + 1.49·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.314049313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314049313\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 23iT - 343T^{2} \) |
| 11 | \( 1 + 30T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 42iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 139T + 6.85e3T^{2} \) |
| 23 | \( 1 + 192iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 234T + 2.43e4T^{2} \) |
| 31 | \( 1 + 55T + 2.97e4T^{2} \) |
| 37 | \( 1 + 191iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 138T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 366iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 330iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 396T + 2.05e5T^{2} \) |
| 61 | \( 1 - 23T + 2.26e5T^{2} \) |
| 67 | \( 1 + 452iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 204T + 3.57e5T^{2} \) |
| 73 | \( 1 + 691iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 709T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.09e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 816T + 7.04e5T^{2} \) |
| 97 | \( 1 + 905iT - 9.12e5T^{2} \) |
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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
−8.899284839362431612641363392741, −7.85234797402414427356938183499, −7.41386820279147379520463063780, −6.27217831048045250812603944092, −5.09885397962351198421637039525, −4.49594481256054279876016463412, −3.32344450262716602473195687788, −2.63738695923539061970066372381, −1.09067387698420127719921167803, −0.34720063351703989519013264430,
1.40956869452361970055702115564, 2.69162723651707337127576603747, 3.69487395534410949852192498425, 5.09589496011161255403879044032, 5.42332533482603929568606379003, 6.38842106188074351063281374189, 7.27971347753815943053322472452, 8.099994957858533972928547740370, 8.771931785847131266481168715539, 9.570481969006266341549154790372
