| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 11-s + 14-s − 16-s − 2·17-s + 4·19-s − 22-s − 4·23-s − 5·25-s − 28-s + 4·29-s − 4·31-s + 5·32-s − 2·34-s + 8·37-s + 4·38-s + 10·41-s + 44-s − 4·46-s − 2·47-s + 49-s − 5·50-s − 6·53-s − 3·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 0.301·11-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.213·22-s − 0.834·23-s − 25-s − 0.188·28-s + 0.742·29-s − 0.718·31-s + 0.883·32-s − 0.342·34-s + 1.31·37-s + 0.648·38-s + 1.56·41-s + 0.150·44-s − 0.589·46-s − 0.291·47-s + 1/7·49-s − 0.707·50-s − 0.824·53-s − 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.392795279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.392795279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.64124142065636, −13.08547217337632, −12.79566557894001, −12.20939408092622, −11.67367428894631, −11.37203687826204, −10.79190077141244, −10.04405750315966, −9.697675284356327, −9.254864272971975, −8.650663075327438, −8.112253982648471, −7.705507070146804, −7.163656622411328, −6.244728652206872, −6.075189258091702, −5.357207318779566, −4.982903842019705, −4.366952360230255, −3.887736788479490, −3.419033445786871, −2.561924157643120, −2.201638014836195, −1.162865220998867, −0.4585706369482276,
0.4585706369482276, 1.162865220998867, 2.201638014836195, 2.561924157643120, 3.419033445786871, 3.887736788479490, 4.366952360230255, 4.982903842019705, 5.357207318779566, 6.075189258091702, 6.244728652206872, 7.163656622411328, 7.705507070146804, 8.112253982648471, 8.650663075327438, 9.254864272971975, 9.697675284356327, 10.04405750315966, 10.79190077141244, 11.37203687826204, 11.67367428894631, 12.20939408092622, 12.79566557894001, 13.08547217337632, 13.64124142065636
