| L(s) = 1 | − 5-s − 5·7-s − 13-s − 6·17-s − 5·19-s − 9·23-s + 25-s + 4·31-s + 5·35-s − 10·37-s + 3·41-s − 8·43-s − 3·47-s + 18·49-s − 3·53-s + 9·59-s + 8·61-s + 65-s + 4·67-s + 6·71-s + 2·73-s − 2·79-s + 6·83-s + 6·85-s − 6·89-s + 5·91-s + 5·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.88·7-s − 0.277·13-s − 1.45·17-s − 1.14·19-s − 1.87·23-s + 1/5·25-s + 0.718·31-s + 0.845·35-s − 1.64·37-s + 0.468·41-s − 1.21·43-s − 0.437·47-s + 18/7·49-s − 0.412·53-s + 1.17·59-s + 1.02·61-s + 0.124·65-s + 0.488·67-s + 0.712·71-s + 0.234·73-s − 0.225·79-s + 0.658·83-s + 0.650·85-s − 0.635·89-s + 0.524·91-s + 0.512·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2932984118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2932984118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.191436625964720571247233183224, −7.03255612161183974806300491646, −6.63052437510913253356243344898, −6.16251661729208258243197155690, −5.17610424662319103588929848331, −4.11286542146601995401404699165, −3.76580342434149920864922732790, −2.75821095813930530563749280441, −2.02911766013951221285234106989, −0.25805400859275701533342521657,
0.25805400859275701533342521657, 2.02911766013951221285234106989, 2.75821095813930530563749280441, 3.76580342434149920864922732790, 4.11286542146601995401404699165, 5.17610424662319103588929848331, 6.16251661729208258243197155690, 6.63052437510913253356243344898, 7.03255612161183974806300491646, 8.191436625964720571247233183224
