L(s) = 1 | + 0.739·3-s + 2.08·5-s + 4.12·7-s − 2.45·9-s − 4.72·11-s − 3.93·13-s + 1.53·15-s + 1.15·17-s + 19-s + 3.04·21-s − 5.67·23-s − 0.661·25-s − 4.03·27-s − 8.45·29-s − 2.39·31-s − 3.49·33-s + 8.58·35-s + 5.34·37-s − 2.90·39-s + 0.419·41-s + 8.48·43-s − 5.11·45-s − 7.51·47-s + 9.98·49-s + 0.851·51-s − 6.91·53-s − 9.84·55-s + ⋯ |
L(s) = 1 | + 0.426·3-s + 0.931·5-s + 1.55·7-s − 0.817·9-s − 1.42·11-s − 1.09·13-s + 0.397·15-s + 0.279·17-s + 0.229·19-s + 0.664·21-s − 1.18·23-s − 0.132·25-s − 0.775·27-s − 1.56·29-s − 0.430·31-s − 0.608·33-s + 1.45·35-s + 0.878·37-s − 0.465·39-s + 0.0655·41-s + 1.29·43-s − 0.761·45-s − 1.09·47-s + 1.42·49-s + 0.119·51-s − 0.950·53-s − 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 - 0.739T + 3T^{2} \) |
| 5 | \( 1 - 2.08T + 5T^{2} \) |
| 7 | \( 1 - 4.12T + 7T^{2} \) |
| 11 | \( 1 + 4.72T + 11T^{2} \) |
| 13 | \( 1 + 3.93T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 23 | \( 1 + 5.67T + 23T^{2} \) |
| 29 | \( 1 + 8.45T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 - 0.419T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 7.51T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 1.82T + 59T^{2} \) |
| 61 | \( 1 + 0.433T + 61T^{2} \) |
| 67 | \( 1 + 9.55T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 3.54T + 73T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.85148341713440617251113930483, −7.38022103980318729482299754949, −5.99767263184767369501881696858, −5.52830371251347288954982680768, −5.03036414112675898836113341567, −4.15538785822327056925164514056, −2.92748650766118346903140588473, −2.24841594780617887954683263785, −1.70741108633124651966534340834, 0,
1.70741108633124651966534340834, 2.24841594780617887954683263785, 2.92748650766118346903140588473, 4.15538785822327056925164514056, 5.03036414112675898836113341567, 5.52830371251347288954982680768, 5.99767263184767369501881696858, 7.38022103980318729482299754949, 7.85148341713440617251113930483