L(s) = 1 | − 0.414·2-s + 1.41·3-s − 1.82·4-s + 5-s − 0.585·6-s + 1.58·8-s − 0.999·9-s − 0.414·10-s + 11-s − 2.58·12-s − 3.41·13-s + 1.41·15-s + 3·16-s + 0.585·17-s + 0.414·18-s − 1.82·20-s − 0.414·22-s + 8.82·23-s + 2.24·24-s + 25-s + 1.41·26-s − 5.65·27-s − 0.828·29-s − 0.585·30-s + 1.75·31-s − 4.41·32-s + 1.41·33-s + ⋯ |
L(s) = 1 | − 0.292·2-s + 0.816·3-s − 0.914·4-s + 0.447·5-s − 0.239·6-s + 0.560·8-s − 0.333·9-s − 0.130·10-s + 0.301·11-s − 0.746·12-s − 0.946·13-s + 0.365·15-s + 0.750·16-s + 0.142·17-s + 0.0976·18-s − 0.408·20-s − 0.0883·22-s + 1.84·23-s + 0.457·24-s + 0.200·25-s + 0.277·26-s − 1.08·27-s − 0.153·29-s − 0.106·30-s + 0.315·31-s − 0.780·32-s + 0.246·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.679218862\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.679218862\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 0.585T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 0.828T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 1.17T + 37T^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 7.89T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 6.48T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 - 5.07T + 73T^{2} \) |
| 79 | \( 1 - 6.48T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 0.828T + 89T^{2} \) |
| 97 | \( 1 + 9.31T + 97T^{2} \) |
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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
−8.814873933362081416538547100811, −8.350557892838430088837210616136, −7.49171599927756629281356598874, −6.75074227383924411010401592858, −5.51921677549521514827465903196, −4.99974065910281688519702751434, −3.97792169777703865695293780784, −3.08746976654748508842946461173, −2.18552752386762504089161215281, −0.826143074917991780636623913178,
0.826143074917991780636623913178, 2.18552752386762504089161215281, 3.08746976654748508842946461173, 3.97792169777703865695293780784, 4.99974065910281688519702751434, 5.51921677549521514827465903196, 6.75074227383924411010401592858, 7.49171599927756629281356598874, 8.350557892838430088837210616136, 8.814873933362081416538547100811