| L(s) = 1 | − 5·5-s − 4·7-s − 40·11-s − 90·13-s + 70·17-s + 40·19-s − 108·23-s + 25·25-s − 166·29-s − 40·31-s + 20·35-s − 130·37-s + 310·41-s − 268·43-s + 556·47-s − 327·49-s + 370·53-s + 200·55-s − 240·59-s − 130·61-s + 450·65-s + 876·67-s + 840·71-s + 250·73-s + 160·77-s − 880·79-s + 188·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.215·7-s − 1.09·11-s − 1.92·13-s + 0.998·17-s + 0.482·19-s − 0.979·23-s + 1/5·25-s − 1.06·29-s − 0.231·31-s + 0.0965·35-s − 0.577·37-s + 1.18·41-s − 0.950·43-s + 1.72·47-s − 0.953·49-s + 0.958·53-s + 0.490·55-s − 0.529·59-s − 0.272·61-s + 0.858·65-s + 1.59·67-s + 1.40·71-s + 0.400·73-s + 0.236·77-s − 1.25·79-s + 0.248·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9455140507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9455140507\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 7 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 90 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 40 T + p^{3} T^{2} \) |
| 23 | \( 1 + 108 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + 40 T + p^{3} T^{2} \) |
| 37 | \( 1 + 130 T + p^{3} T^{2} \) |
| 41 | \( 1 - 310 T + p^{3} T^{2} \) |
| 43 | \( 1 + 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 556 T + p^{3} T^{2} \) |
| 53 | \( 1 - 370 T + p^{3} T^{2} \) |
| 59 | \( 1 + 240 T + p^{3} T^{2} \) |
| 61 | \( 1 + 130 T + p^{3} T^{2} \) |
| 67 | \( 1 - 876 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 250 T + p^{3} T^{2} \) |
| 79 | \( 1 + 880 T + p^{3} T^{2} \) |
| 83 | \( 1 - 188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1550 T + p^{3} T^{2} \) |
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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
−9.355927509627335852473984556907, −8.086919121630046029862346378856, −7.62668597849544585483371635958, −6.96692062986203239792854121155, −5.62248139232690598008883731919, −5.12646340000089761802342101097, −4.03089947384815292088439117493, −2.99388485233309469256197650230, −2.10796208857406477375355135679, −0.45362168349166410274509632703,
0.45362168349166410274509632703, 2.10796208857406477375355135679, 2.99388485233309469256197650230, 4.03089947384815292088439117493, 5.12646340000089761802342101097, 5.62248139232690598008883731919, 6.96692062986203239792854121155, 7.62668597849544585483371635958, 8.086919121630046029862346378856, 9.355927509627335852473984556907
