L(s) = 1 | − 1.38·3-s − 1.32·5-s + 2.39·7-s − 1.08·9-s + 3.31·11-s − 3.52·13-s + 1.82·15-s + 2.72·17-s + 6.86·19-s − 3.31·21-s + 6.61·23-s − 3.25·25-s + 5.65·27-s − 7.66·29-s − 1.56·31-s − 4.58·33-s − 3.16·35-s + 1.61·37-s + 4.88·39-s + 3.10·41-s − 2.57·43-s + 1.43·45-s − 2.66·47-s − 1.26·49-s − 3.77·51-s + 3.91·53-s − 4.37·55-s + ⋯ |
L(s) = 1 | − 0.799·3-s − 0.590·5-s + 0.905·7-s − 0.360·9-s + 0.999·11-s − 0.978·13-s + 0.472·15-s + 0.660·17-s + 1.57·19-s − 0.723·21-s + 1.37·23-s − 0.650·25-s + 1.08·27-s − 1.42·29-s − 0.281·31-s − 0.798·33-s − 0.535·35-s + 0.265·37-s + 0.782·39-s + 0.484·41-s − 0.393·43-s + 0.213·45-s − 0.388·47-s − 0.180·49-s − 0.528·51-s + 0.537·53-s − 0.590·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.374016969\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.374016969\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 + 1.38T + 3T^{2} \) |
| 5 | \( 1 + 1.32T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 13 | \( 1 + 3.52T + 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 - 6.86T + 19T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 - 3.10T + 41T^{2} \) |
| 43 | \( 1 + 2.57T + 43T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 - 3.91T + 53T^{2} \) |
| 59 | \( 1 - 1.25T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 9.01T + 67T^{2} \) |
| 71 | \( 1 - 6.20T + 71T^{2} \) |
| 73 | \( 1 + 8.76T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 3.05T + 97T^{2} \) |
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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
−7.944678562783609617124261272694, −7.31993546404641460755628105745, −6.82969225944639613774487147841, −5.62127941884277790451864660975, −5.37684386148589262933688021397, −4.58021195118303924095595776028, −3.71884456181469286983297367501, −2.88247044376578693535567575078, −1.61198759660071970950316271781, −0.67444187286535653805058292090,
0.67444187286535653805058292090, 1.61198759660071970950316271781, 2.88247044376578693535567575078, 3.71884456181469286983297367501, 4.58021195118303924095595776028, 5.37684386148589262933688021397, 5.62127941884277790451864660975, 6.82969225944639613774487147841, 7.31993546404641460755628105745, 7.944678562783609617124261272694