| L(s) = 1 | − 1.82·2-s + 1.32·4-s − 2.42·7-s + 1.22·8-s + 0.0262·11-s − 2.72·13-s + 4.43·14-s − 4.89·16-s + 4.47·17-s + 0.925·19-s − 0.0478·22-s + 9.37·23-s + 4.96·26-s − 3.22·28-s − 5.41·29-s − 2.25·31-s + 6.47·32-s − 8.15·34-s + 37-s − 1.68·38-s − 8.04·41-s − 8.72·43-s + 0.0348·44-s − 17.0·46-s − 4.16·47-s − 1.09·49-s − 3.61·52-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 0.663·4-s − 0.918·7-s + 0.433·8-s + 0.00790·11-s − 0.755·13-s + 1.18·14-s − 1.22·16-s + 1.08·17-s + 0.212·19-s − 0.0101·22-s + 1.95·23-s + 0.974·26-s − 0.609·28-s − 1.00·29-s − 0.404·31-s + 1.14·32-s − 1.39·34-s + 0.164·37-s − 0.274·38-s − 1.25·41-s − 1.33·43-s + 0.00524·44-s − 2.52·46-s − 0.607·47-s − 0.156·49-s − 0.501·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 7 | \( 1 + 2.42T + 7T^{2} \) |
| 11 | \( 1 - 0.0262T + 11T^{2} \) |
| 13 | \( 1 + 2.72T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 0.925T + 19T^{2} \) |
| 23 | \( 1 - 9.37T + 23T^{2} \) |
| 29 | \( 1 + 5.41T + 29T^{2} \) |
| 31 | \( 1 + 2.25T + 31T^{2} \) |
| 41 | \( 1 + 8.04T + 41T^{2} \) |
| 43 | \( 1 + 8.72T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 - 7.22T + 59T^{2} \) |
| 61 | \( 1 + 8.71T + 61T^{2} \) |
| 67 | \( 1 - 8.78T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 2.40T + 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 + 8.56T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 - 8.43T + 97T^{2} \) |
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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
−7.38359495341972285162876856865, −7.13384909311410066278032830024, −6.36599158713149074396879217162, −5.33429381277546108338692618988, −4.83064447305985967486977083786, −3.61606538314816852088420763580, −3.02462640105069113649982357729, −1.95776750589645900212894313142, −0.990057337014672658386059895214, 0,
0.990057337014672658386059895214, 1.95776750589645900212894313142, 3.02462640105069113649982357729, 3.61606538314816852088420763580, 4.83064447305985967486977083786, 5.33429381277546108338692618988, 6.36599158713149074396879217162, 7.13384909311410066278032830024, 7.38359495341972285162876856865
