L(s) = 1 | − 0.319·3-s + 2.22·5-s + 0.484·7-s − 2.89·9-s − 3.07·11-s − 5.65·13-s − 0.710·15-s + 1.35·17-s − 2.21·19-s − 0.154·21-s + 1.53·23-s − 0.0565·25-s + 1.88·27-s − 3.86·29-s + 5.33·31-s + 0.982·33-s + 1.07·35-s + 6.61·37-s + 1.80·39-s − 0.811·41-s + 0.404·43-s − 6.44·45-s − 47-s − 6.76·49-s − 0.432·51-s + 9.23·53-s − 6.83·55-s + ⋯ |
L(s) = 1 | − 0.184·3-s + 0.994·5-s + 0.183·7-s − 0.965·9-s − 0.926·11-s − 1.56·13-s − 0.183·15-s + 0.328·17-s − 0.509·19-s − 0.0337·21-s + 0.319·23-s − 0.0113·25-s + 0.362·27-s − 0.717·29-s + 0.957·31-s + 0.170·33-s + 0.182·35-s + 1.08·37-s + 0.289·39-s − 0.126·41-s + 0.0616·43-s − 0.960·45-s − 0.145·47-s − 0.966·49-s − 0.0606·51-s + 1.26·53-s − 0.921·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.522759586\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.522759586\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 47 | \( 1 + T \) |
good | 3 | \( 1 + 0.319T + 3T^{2} \) |
| 5 | \( 1 - 2.22T + 5T^{2} \) |
| 7 | \( 1 - 0.484T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 3.86T + 29T^{2} \) |
| 31 | \( 1 - 5.33T + 31T^{2} \) |
| 37 | \( 1 - 6.61T + 37T^{2} \) |
| 41 | \( 1 + 0.811T + 41T^{2} \) |
| 43 | \( 1 - 0.404T + 43T^{2} \) |
| 53 | \( 1 - 9.23T + 53T^{2} \) |
| 59 | \( 1 - 1.47T + 59T^{2} \) |
| 61 | \( 1 - 2.46T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 3.68T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−8.019023906803399317311743764011, −7.46716486058115196539371016301, −6.50545239612849589813590466560, −5.90980233626355027303446529827, −5.13993416463790579313513128263, −4.84245051863510595905867346126, −3.51598217855072992614994799372, −2.43691234487269414100794745580, −2.22558357669153795086935429850, −0.61884588254630013703553648753,
0.61884588254630013703553648753, 2.22558357669153795086935429850, 2.43691234487269414100794745580, 3.51598217855072992614994799372, 4.84245051863510595905867346126, 5.13993416463790579313513128263, 5.90980233626355027303446529827, 6.50545239612849589813590466560, 7.46716486058115196539371016301, 8.019023906803399317311743764011