| L(s) = 1 | − 4·7-s − 11-s − 13-s + 4·19-s − 5·25-s − 10·31-s − 2·37-s + 6·41-s + 10·43-s + 9·49-s + 6·53-s − 2·61-s − 2·67-s − 12·71-s − 10·73-s + 4·77-s − 10·79-s − 12·83-s − 12·89-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.301·11-s − 0.277·13-s + 0.917·19-s − 25-s − 1.79·31-s − 0.328·37-s + 0.937·41-s + 1.52·43-s + 9/7·49-s + 0.824·53-s − 0.256·61-s − 0.244·67-s − 1.42·71-s − 1.17·73-s + 0.455·77-s − 1.12·79-s − 1.31·83-s − 1.27·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6187159313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6187159313\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.92793394228214, −13.32161231633613, −13.05479871743788, −12.45933291651807, −12.16494928556554, −11.47261821756928, −10.97472799245945, −10.38863798449097, −9.868970701539788, −9.512351797697454, −9.059669926736009, −8.518916839384606, −7.677543451956547, −7.230883178190115, −7.017353148764175, −6.028714472290658, −5.778987007369337, −5.350648829461244, −4.301850712304676, −3.995238408687453, −3.174514211459937, −2.862184442211698, −2.095104215579476, −1.244312140120023, −0.2605805439008605,
0.2605805439008605, 1.244312140120023, 2.095104215579476, 2.862184442211698, 3.174514211459937, 3.995238408687453, 4.301850712304676, 5.350648829461244, 5.778987007369337, 6.028714472290658, 7.017353148764175, 7.230883178190115, 7.677543451956547, 8.518916839384606, 9.059669926736009, 9.512351797697454, 9.868970701539788, 10.38863798449097, 10.97472799245945, 11.47261821756928, 12.16494928556554, 12.45933291651807, 13.05479871743788, 13.32161231633613, 13.92793394228214
