L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s + 2·11-s − 2·13-s − 15-s + 2·17-s + 2·19-s + 4·21-s + 25-s − 27-s − 2·29-s + 4·31-s − 2·33-s − 4·35-s + 2·39-s − 6·41-s − 4·43-s + 45-s + 10·47-s + 9·49-s − 2·51-s + 6·53-s + 2·55-s − 2·57-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.458·19-s + 0.872·21-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s − 0.676·35-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.45·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.269·55-s − 0.264·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.70917342884763, −12.38655442546436, −11.94940542813459, −11.64085023220991, −10.97830813170156, −10.43934777632535, −9.997357622808559, −9.740581690660442, −9.398778900099200, −8.743576807829269, −8.390825381938250, −7.525371548130183, −7.110254113102897, −6.821437038258621, −6.259807733190348, −5.781224460945501, −5.541857572654890, −4.780995667002393, −4.357189787884462, −3.499721364988937, −3.411559854709191, −2.603624975458028, −2.112948330379209, −1.268824211349045, −0.7219670577241895, 0,
0.7219670577241895, 1.268824211349045, 2.112948330379209, 2.603624975458028, 3.411559854709191, 3.499721364988937, 4.357189787884462, 4.780995667002393, 5.541857572654890, 5.781224460945501, 6.259807733190348, 6.821437038258621, 7.110254113102897, 7.525371548130183, 8.390825381938250, 8.743576807829269, 9.398778900099200, 9.740581690660442, 9.997357622808559, 10.43934777632535, 10.97830813170156, 11.64085023220991, 11.94940542813459, 12.38655442546436, 12.70917342884763