| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s + 2·13-s + 15-s + 6·17-s + 4·19-s − 4·21-s + 25-s − 27-s − 6·29-s − 8·31-s − 4·35-s + 2·37-s − 2·39-s − 6·41-s + 4·43-s − 45-s + 9·49-s − 6·51-s − 6·53-s − 4·57-s − 10·61-s + 4·63-s − 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 0.529·57-s − 1.28·61-s + 0.503·63-s − 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.158392111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158392111\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−11.85503029140590792221076481507, −11.32300197915863352372468915281, −10.48322857979944854900497529739, −9.206120678796429021467253404807, −7.965215938754459889106897012614, −7.38482725826507043429631860666, −5.76555679083117871916592673419, −4.95303891523898796353167575120, −3.62987230449253956202768748221, −1.44399343799308961081305068602,
1.44399343799308961081305068602, 3.62987230449253956202768748221, 4.95303891523898796353167575120, 5.76555679083117871916592673419, 7.38482725826507043429631860666, 7.965215938754459889106897012614, 9.206120678796429021467253404807, 10.48322857979944854900497529739, 11.32300197915863352372468915281, 11.85503029140590792221076481507
