L(s) = 1 | − 2.73·3-s − 5-s + 4.73·7-s + 4.46·9-s + 3.46·11-s − 3.46·13-s + 2.73·15-s − 3.46·17-s + 2·19-s − 12.9·21-s − 2.19·23-s + 25-s − 3.99·27-s + 2.53·31-s − 9.46·33-s − 4.73·35-s + 6·37-s + 9.46·39-s − 9.46·41-s − 0.196·43-s − 4.46·45-s − 2.19·47-s + 15.3·49-s + 9.46·51-s + 10.3·53-s − 3.46·55-s − 5.46·57-s + ⋯ |
L(s) = 1 | − 1.57·3-s − 0.447·5-s + 1.78·7-s + 1.48·9-s + 1.04·11-s − 0.960·13-s + 0.705·15-s − 0.840·17-s + 0.458·19-s − 2.82·21-s − 0.457·23-s + 0.200·25-s − 0.769·27-s + 0.455·31-s − 1.64·33-s − 0.799·35-s + 0.986·37-s + 1.51·39-s − 1.47·41-s − 0.0299·43-s − 0.665·45-s − 0.320·47-s + 2.19·49-s + 1.32·51-s + 1.42·53-s − 0.467·55-s − 0.723·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.049195289\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.049195289\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 + 0.196T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 0.928T + 61T^{2} \) |
| 67 | \( 1 - 0.196T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 1.26T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.897584006791258754958413460763, −8.793071638074079137227094806959, −7.916359515231302360463770440660, −7.09902113784128903080692335842, −6.34974248156783198313216043871, −5.25120503777538650817392480543, −4.77445186113098724680085868514, −4.00599210164180113253134503962, −2.04512602556676338221682428014, −0.847842980116801684385043312983,
0.847842980116801684385043312983, 2.04512602556676338221682428014, 4.00599210164180113253134503962, 4.77445186113098724680085868514, 5.25120503777538650817392480543, 6.34974248156783198313216043871, 7.09902113784128903080692335842, 7.916359515231302360463770440660, 8.793071638074079137227094806959, 9.897584006791258754958413460763