| L(s) = 1 | − 1.18·2-s − 0.595·4-s − 2.75·7-s + 3.07·8-s − 4.94·11-s − 0.730·13-s + 3.26·14-s − 2.45·16-s − 1.08·17-s + 0.275·19-s + 5.85·22-s + 6.53·23-s + 0.865·26-s + 1.64·28-s − 4.89·29-s − 31-s − 3.24·32-s + 1.28·34-s + 11.2·37-s − 0.326·38-s + 0.301·41-s + 7.63·43-s + 2.94·44-s − 7.74·46-s − 1.07·47-s + 0.595·49-s + 0.434·52-s + ⋯ |
| L(s) = 1 | − 0.838·2-s − 0.297·4-s − 1.04·7-s + 1.08·8-s − 1.48·11-s − 0.202·13-s + 0.872·14-s − 0.613·16-s − 0.262·17-s + 0.0631·19-s + 1.24·22-s + 1.36·23-s + 0.169·26-s + 0.310·28-s − 0.909·29-s − 0.179·31-s − 0.573·32-s + 0.219·34-s + 1.85·37-s − 0.0529·38-s + 0.0470·41-s + 1.16·43-s + 0.443·44-s − 1.14·46-s − 0.156·47-s + 0.0850·49-s + 0.0602·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.18T + 2T^{2} \) |
| 7 | \( 1 + 2.75T + 7T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 + 0.730T + 13T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 - 0.275T + 19T^{2} \) |
| 23 | \( 1 - 6.53T + 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 0.301T + 41T^{2} \) |
| 43 | \( 1 - 7.63T + 43T^{2} \) |
| 47 | \( 1 + 1.07T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 3.62T + 59T^{2} \) |
| 61 | \( 1 + 6.64T + 61T^{2} \) |
| 67 | \( 1 + 7.25T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 + 9.13T + 73T^{2} \) |
| 79 | \( 1 - 9.63T + 79T^{2} \) |
| 83 | \( 1 - 7.71T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 0.305T + 97T^{2} \) |
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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
−7.59239074747912413373388597184, −7.24635622520147925402158950508, −6.27283946821635975107876044486, −5.46907637448336127889398312141, −4.81001214271606873346038274153, −3.96377141943864163667163912573, −2.98909043475114221353611742266, −2.28789629993522987621236891246, −0.913271046515009763307170098145, 0,
0.913271046515009763307170098145, 2.28789629993522987621236891246, 2.98909043475114221353611742266, 3.96377141943864163667163912573, 4.81001214271606873346038274153, 5.46907637448336127889398312141, 6.27283946821635975107876044486, 7.24635622520147925402158950508, 7.59239074747912413373388597184
