L(s) = 1 | + 2.77·2-s + 5.72·4-s − 2.25·5-s + 0.879·7-s + 10.3·8-s − 6.27·10-s + 11-s + 3.02·13-s + 2.44·14-s + 17.2·16-s − 2.27·17-s + 3.93·19-s − 12.9·20-s + 2.77·22-s + 3.56·23-s + 0.0920·25-s + 8.40·26-s + 5.02·28-s + 1.57·29-s − 9.17·31-s + 27.3·32-s − 6.32·34-s − 1.98·35-s − 7.15·37-s + 10.9·38-s − 23.3·40-s + 7.02·41-s + ⋯ |
L(s) = 1 | + 1.96·2-s + 2.86·4-s − 1.00·5-s + 0.332·7-s + 3.65·8-s − 1.98·10-s + 0.301·11-s + 0.838·13-s + 0.652·14-s + 4.32·16-s − 0.552·17-s + 0.903·19-s − 2.88·20-s + 0.592·22-s + 0.743·23-s + 0.0184·25-s + 1.64·26-s + 0.950·28-s + 0.292·29-s − 1.64·31-s + 4.83·32-s − 1.08·34-s − 0.335·35-s − 1.17·37-s + 1.77·38-s − 3.68·40-s + 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.609431119\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.609431119\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 0.879T + 7T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 - 3.93T + 19T^{2} \) |
| 23 | \( 1 - 3.56T + 23T^{2} \) |
| 29 | \( 1 - 1.57T + 29T^{2} \) |
| 31 | \( 1 + 9.17T + 31T^{2} \) |
| 37 | \( 1 + 7.15T + 37T^{2} \) |
| 41 | \( 1 - 7.02T + 41T^{2} \) |
| 43 | \( 1 - 8.40T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 1.97T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 67 | \( 1 - 0.158T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 7.62T + 79T^{2} \) |
| 83 | \( 1 - 8.04T + 83T^{2} \) |
| 89 | \( 1 + 6.83T + 89T^{2} \) |
| 97 | \( 1 + 2.44T + 97T^{2} \) |
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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
−7.66175099856478678973779134676, −7.20311918247183627915128687599, −6.54456372234903449496812021570, −5.68002552912620180201939881610, −5.15886000139292652140130526617, −4.28285438106573131946146063226, −3.80341076913770943693434507620, −3.21278522029486421744923610800, −2.21568499666920742271401183581, −1.17089615182639809424965621164,
1.17089615182639809424965621164, 2.21568499666920742271401183581, 3.21278522029486421744923610800, 3.80341076913770943693434507620, 4.28285438106573131946146063226, 5.15886000139292652140130526617, 5.68002552912620180201939881610, 6.54456372234903449496812021570, 7.20311918247183627915128687599, 7.66175099856478678973779134676