L(s) = 1 | − 5-s − 2·7-s + 4·11-s − 13-s − 2·19-s − 2·23-s + 25-s − 4·29-s − 4·31-s + 2·35-s − 2·37-s + 6·41-s + 4·43-s + 8·47-s − 3·49-s + 2·53-s − 4·55-s + 4·59-s − 2·61-s + 65-s − 8·67-s + 8·71-s − 4·73-s − 8·77-s − 8·79-s + 12·83-s − 6·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.277·13-s − 0.458·19-s − 0.417·23-s + 1/5·25-s − 0.742·29-s − 0.718·31-s + 0.338·35-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.539·55-s + 0.520·59-s − 0.256·61-s + 0.124·65-s − 0.977·67-s + 0.949·71-s − 0.468·73-s − 0.911·77-s − 0.900·79-s + 1.31·83-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.393965143\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.393965143\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.47026781828271321780524861328, −7.16705936520112665196405669007, −6.26576788571283959765643694487, −5.87796143364547794684792520657, −4.85295274538705807525687513031, −4.00551820709711944153946336107, −3.64404969039552150892106194625, −2.66618740544468384122796279111, −1.72379762648236565652317886620, −0.56280110853164977758252165067,
0.56280110853164977758252165067, 1.72379762648236565652317886620, 2.66618740544468384122796279111, 3.64404969039552150892106194625, 4.00551820709711944153946336107, 4.85295274538705807525687513031, 5.87796143364547794684792520657, 6.26576788571283959765643694487, 7.16705936520112665196405669007, 7.47026781828271321780524861328