L(s) = 1 | + 3.00·3-s + 3.90·5-s − 4.05·7-s + 6.02·9-s − 1.15·11-s + 3.06·13-s + 11.7·15-s + 2.39·17-s − 5.08·19-s − 12.1·21-s + 5.68·23-s + 10.2·25-s + 9.07·27-s − 6.19·29-s + 1.20·31-s − 3.47·33-s − 15.8·35-s − 0.702·37-s + 9.19·39-s + 3.20·41-s + 0.852·43-s + 23.4·45-s + 9.40·47-s + 9.43·49-s + 7.18·51-s + 2.44·53-s − 4.51·55-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1.74·5-s − 1.53·7-s + 2.00·9-s − 0.349·11-s + 0.849·13-s + 3.02·15-s + 0.580·17-s − 1.16·19-s − 2.65·21-s + 1.18·23-s + 2.04·25-s + 1.74·27-s − 1.15·29-s + 0.215·31-s − 0.605·33-s − 2.67·35-s − 0.115·37-s + 1.47·39-s + 0.500·41-s + 0.129·43-s + 3.50·45-s + 1.37·47-s + 1.34·49-s + 1.00·51-s + 0.335·53-s − 0.609·55-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
\(L(1)\) |
\(\approx\) |
\(4.960234244\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.960234244\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.00T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 3.06T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 - 1.20T + 31T^{2} \) |
| 37 | \( 1 + 0.702T + 37T^{2} \) |
| 41 | \( 1 - 3.20T + 41T^{2} \) |
| 43 | \( 1 - 0.852T + 43T^{2} \) |
| 47 | \( 1 - 9.40T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 - 8.28T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 0.0237T + 71T^{2} \) |
| 73 | \( 1 - 1.97T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 9.64T + 89T^{2} \) |
| 97 | \( 1 + 4.93T + 97T^{2} \) |
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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
−8.449160659237789296586165306677, −7.24734599305858438396799682687, −6.78774728496251651199097263179, −5.99244713427466072539045949374, −5.41268200739736810459205576984, −4.09548390928185257887720257601, −3.42360251990765594271977096044, −2.65067418029843989180596036081, −2.22080494136285645285517199231, −1.14094793215313288206764184197,
1.14094793215313288206764184197, 2.22080494136285645285517199231, 2.65067418029843989180596036081, 3.42360251990765594271977096044, 4.09548390928185257887720257601, 5.41268200739736810459205576984, 5.99244713427466072539045949374, 6.78774728496251651199097263179, 7.24734599305858438396799682687, 8.449160659237789296586165306677