| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 5·11-s − 13-s + 15-s − 6·17-s + 19-s + 21-s − 23-s − 4·25-s + 5·27-s + 6·29-s − 5·33-s + 35-s − 6·37-s + 39-s − 6·41-s + 10·43-s + 2·45-s + 49-s + 6·51-s + 8·53-s − 5·55-s − 57-s − 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.229·19-s + 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.962·27-s + 1.11·29-s − 0.870·33-s + 0.169·35-s − 0.986·37-s + 0.160·39-s − 0.937·41-s + 1.52·43-s + 0.298·45-s + 1/7·49-s + 0.840·51-s + 1.09·53-s − 0.674·55-s − 0.132·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4573232891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4573232891\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.53423256222098, −12.90751650133640, −12.15974217637418, −12.10693552711712, −11.47428879269889, −11.33830409345552, −10.44644720230040, −10.34061656838703, −9.440029483876394, −9.053180312224289, −8.691852274914425, −8.167593128558488, −7.420758496099927, −6.871372222058621, −6.591159693260224, −5.949433393137661, −5.637248129565561, −4.776632755114783, −4.308434907143107, −3.926587561726203, −3.139877966998175, −2.655637303919909, −1.823180376723985, −1.138286863841574, −0.2264059664078957,
0.2264059664078957, 1.138286863841574, 1.823180376723985, 2.655637303919909, 3.139877966998175, 3.926587561726203, 4.308434907143107, 4.776632755114783, 5.637248129565561, 5.949433393137661, 6.591159693260224, 6.871372222058621, 7.420758496099927, 8.167593128558488, 8.691852274914425, 9.053180312224289, 9.440029483876394, 10.34061656838703, 10.44644720230040, 11.33830409345552, 11.47428879269889, 12.10693552711712, 12.15974217637418, 12.90751650133640, 13.53423256222098
