| L(s) = 1 | − 3-s − 3·7-s + 9-s − 4·11-s + 13-s + 7·19-s + 3·21-s − 6·23-s − 27-s − 29-s + 7·31-s + 4·33-s − 2·37-s − 39-s − 5·41-s − 43-s − 2·47-s + 2·49-s − 10·53-s − 7·57-s − 2·59-s − 61-s − 3·63-s + 5·67-s + 6·69-s + 8·71-s + 7·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.60·19-s + 0.654·21-s − 1.25·23-s − 0.192·27-s − 0.185·29-s + 1.25·31-s + 0.696·33-s − 0.328·37-s − 0.160·39-s − 0.780·41-s − 0.152·43-s − 0.291·47-s + 2/7·49-s − 1.37·53-s − 0.927·57-s − 0.260·59-s − 0.128·61-s − 0.377·63-s + 0.610·67-s + 0.722·69-s + 0.949·71-s + 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.18214663283692, −12.79174461743637, −12.36471305487263, −11.86536090686155, −11.44478125591133, −10.97587548550192, −10.23412264006518, −10.09277227972885, −9.653503710538564, −9.226081081709412, −8.306266823114112, −8.142228818974177, −7.466236303831613, −7.032194260735168, −6.433391353993343, −6.064390023031774, −5.507345502126816, −5.113404241184814, −4.519408955705272, −3.862328779959292, −3.145581966351284, −3.002334465153501, −2.132252091690423, −1.433357788492285, −0.6076022316627691, 0,
0.6076022316627691, 1.433357788492285, 2.132252091690423, 3.002334465153501, 3.145581966351284, 3.862328779959292, 4.519408955705272, 5.113404241184814, 5.507345502126816, 6.064390023031774, 6.433391353993343, 7.032194260735168, 7.466236303831613, 8.142228818974177, 8.306266823114112, 9.226081081709412, 9.653503710538564, 10.09277227972885, 10.23412264006518, 10.97587548550192, 11.44478125591133, 11.86536090686155, 12.36471305487263, 12.79174461743637, 13.18214663283692
