| L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s − 13-s + 14-s + 16-s − 4·17-s + 2·20-s − 25-s + 26-s − 28-s + 2·29-s + 31-s − 32-s + 4·34-s − 2·35-s + 10·37-s − 2·40-s − 2·41-s − 4·43-s − 2·47-s − 6·49-s + 50-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s + 0.447·20-s − 1/5·25-s + 0.196·26-s − 0.188·28-s + 0.371·29-s + 0.179·31-s − 0.176·32-s + 0.685·34-s − 0.338·35-s + 1.64·37-s − 0.316·40-s − 0.312·41-s − 0.609·43-s − 0.291·47-s − 6/7·49-s + 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−7.83643244692097070875102082356, −6.75682215039475281298086400299, −6.49269170826280172028271747627, −5.69331076627497947363121845927, −4.89234558040917615668229395266, −3.96398837723704926719529253044, −2.86258561925524542045997959803, −2.23847861173910136830326931920, −1.30812330580582401065920757644, 0,
1.30812330580582401065920757644, 2.23847861173910136830326931920, 2.86258561925524542045997959803, 3.96398837723704926719529253044, 4.89234558040917615668229395266, 5.69331076627497947363121845927, 6.49269170826280172028271747627, 6.75682215039475281298086400299, 7.83643244692097070875102082356
