L(s) = 1 | + 2.39·3-s − 2.03·5-s + 4.56·7-s + 2.72·9-s − 1.62·11-s − 13-s − 4.85·15-s + 7.80·17-s + 6.68·19-s + 10.9·21-s − 5.09·23-s − 0.877·25-s − 0.652·27-s + 29-s − 3.04·31-s − 3.88·33-s − 9.26·35-s − 0.865·37-s − 2.39·39-s + 4.42·41-s + 5.81·43-s − 5.53·45-s − 3.77·47-s + 13.8·49-s + 18.6·51-s + 11.2·53-s + 3.29·55-s + ⋯ |
L(s) = 1 | + 1.38·3-s − 0.907·5-s + 1.72·7-s + 0.909·9-s − 0.489·11-s − 0.277·13-s − 1.25·15-s + 1.89·17-s + 1.53·19-s + 2.38·21-s − 1.06·23-s − 0.175·25-s − 0.125·27-s + 0.185·29-s − 0.546·31-s − 0.675·33-s − 1.56·35-s − 0.142·37-s − 0.383·39-s + 0.690·41-s + 0.886·43-s − 0.825·45-s − 0.551·47-s + 1.97·49-s + 2.61·51-s + 1.55·53-s + 0.444·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.687723119\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.687723119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 2.39T + 3T^{2} \) |
| 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 1.62T + 11T^{2} \) |
| 17 | \( 1 - 7.80T + 17T^{2} \) |
| 19 | \( 1 - 6.68T + 19T^{2} \) |
| 23 | \( 1 + 5.09T + 23T^{2} \) |
| 31 | \( 1 + 3.04T + 31T^{2} \) |
| 37 | \( 1 + 0.865T + 37T^{2} \) |
| 41 | \( 1 - 4.42T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 + 3.77T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 - 4.30T + 61T^{2} \) |
| 67 | \( 1 - 8.09T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 - 0.476T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 0.448T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 1.41T + 97T^{2} \) |
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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
−7.979434636133835655661513638137, −7.68902304590317839684911658571, −7.25146953372270034141537921525, −5.61566586821612105765525071459, −5.22827969889622098369266739137, −4.17544537014161620269989096944, −3.65639425936537447933852705254, −2.82369767424440414068373172965, −1.95947320555973759609905244349, −1.00181554577237315713068054641,
1.00181554577237315713068054641, 1.95947320555973759609905244349, 2.82369767424440414068373172965, 3.65639425936537447933852705254, 4.17544537014161620269989096944, 5.22827969889622098369266739137, 5.61566586821612105765525071459, 7.25146953372270034141537921525, 7.68902304590317839684911658571, 7.979434636133835655661513638137