| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 11-s + 13-s + 15-s − 17-s − 19-s − 4·21-s − 6·23-s + 25-s − 27-s − 6·31-s + 33-s − 4·35-s + 11·37-s − 39-s + 11·41-s + 5·43-s − 45-s − 47-s + 9·49-s + 51-s + 12·53-s + 55-s + 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.242·17-s − 0.229·19-s − 0.872·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.07·31-s + 0.174·33-s − 0.676·35-s + 1.80·37-s − 0.160·39-s + 1.71·41-s + 0.762·43-s − 0.149·45-s − 0.145·47-s + 9/7·49-s + 0.140·51-s + 1.64·53-s + 0.134·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.68459228173909, −13.07282648964992, −12.69899307715249, −12.14713111434349, −11.57700714977904, −11.32137752680818, −10.96431087337787, −10.42546443000182, −9.961976448183623, −9.228515539030099, −8.781184699327740, −8.156408125766043, −7.806558789374983, −7.423523794045411, −6.824209395378164, −6.077565542040338, −5.644677357113257, −5.255725351002384, −4.471317829550411, −4.162354384657410, −3.787045956747067, −2.586083026787373, −2.294082398489712, −1.428827206011377, −0.8918201954354565, 0,
0.8918201954354565, 1.428827206011377, 2.294082398489712, 2.586083026787373, 3.787045956747067, 4.162354384657410, 4.471317829550411, 5.255725351002384, 5.644677357113257, 6.077565542040338, 6.824209395378164, 7.423523794045411, 7.806558789374983, 8.156408125766043, 8.781184699327740, 9.228515539030099, 9.961976448183623, 10.42546443000182, 10.96431087337787, 11.32137752680818, 11.57700714977904, 12.14713111434349, 12.69899307715249, 13.07282648964992, 13.68459228173909
