| L(s) = 1 | + 3-s − 3.60·5-s − 4.85·7-s + 9-s + 0.783·11-s − 2.28·13-s − 3.60·15-s − 3.64·17-s − 1.90·19-s − 4.85·21-s + 0.781·23-s + 7.96·25-s + 27-s − 5.79·29-s − 5.38·31-s + 0.783·33-s + 17.4·35-s − 7.39·37-s − 2.28·39-s + 1.19·41-s + 2.44·43-s − 3.60·45-s − 0.0708·47-s + 16.5·49-s − 3.64·51-s + 2.21·53-s − 2.82·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.61·5-s − 1.83·7-s + 0.333·9-s + 0.236·11-s − 0.634·13-s − 0.929·15-s − 0.883·17-s − 0.436·19-s − 1.06·21-s + 0.163·23-s + 1.59·25-s + 0.192·27-s − 1.07·29-s − 0.967·31-s + 0.136·33-s + 2.95·35-s − 1.21·37-s − 0.366·39-s + 0.185·41-s + 0.373·43-s − 0.536·45-s − 0.0103·47-s + 2.37·49-s − 0.510·51-s + 0.303·53-s − 0.380·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5555706268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5555706268\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 3.60T + 5T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 11 | \( 1 - 0.783T + 11T^{2} \) |
| 13 | \( 1 + 2.28T + 13T^{2} \) |
| 17 | \( 1 + 3.64T + 17T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 - 0.781T + 23T^{2} \) |
| 29 | \( 1 + 5.79T + 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 + 7.39T + 37T^{2} \) |
| 41 | \( 1 - 1.19T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 + 0.0708T + 47T^{2} \) |
| 53 | \( 1 - 2.21T + 53T^{2} \) |
| 59 | \( 1 + 3.50T + 59T^{2} \) |
| 61 | \( 1 - 15.0T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 + 6.82T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 0.933T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−8.535794759154474734746797703878, −7.60211951147984988600706082873, −7.04479008530907016712254321591, −6.58766621572353744520522560037, −5.45306174457093056693886310456, −4.30404071045785879287520515853, −3.74437684345010449441867837048, −3.19614715935579330221271424939, −2.22505839913084738996562231978, −0.38660863364376812497496119379,
0.38660863364376812497496119379, 2.22505839913084738996562231978, 3.19614715935579330221271424939, 3.74437684345010449441867837048, 4.30404071045785879287520515853, 5.45306174457093056693886310456, 6.58766621572353744520522560037, 7.04479008530907016712254321591, 7.60211951147984988600706082873, 8.535794759154474734746797703878
