| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 2·11-s + 5·13-s − 15-s + 4·17-s − 6·19-s − 4·21-s + 6·23-s + 25-s − 27-s − 29-s + 4·31-s − 2·33-s + 4·35-s − 5·39-s + 8·41-s − 9·43-s + 45-s + 3·47-s + 9·49-s − 4·51-s + 11·53-s + 2·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s + 1.38·13-s − 0.258·15-s + 0.970·17-s − 1.37·19-s − 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.185·29-s + 0.718·31-s − 0.348·33-s + 0.676·35-s − 0.800·39-s + 1.24·41-s − 1.37·43-s + 0.149·45-s + 0.437·47-s + 9/7·49-s − 0.560·51-s + 1.51·53-s + 0.269·55-s + 0.794·57-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
| \(L(1)\) |
\(\approx\) |
\(4.856684133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.856684133\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.70860690386981, −11.92334840769550, −11.70852996612901, −11.21482109690835, −10.77508112618093, −10.58968900879627, −9.941624020861946, −9.435783113212823, −8.746070213711872, −8.513557264974927, −8.162364890010415, −7.474760676397385, −6.940955647917305, −6.545591268754347, −5.966402964652255, −5.509376585172712, −5.217103447834972, −4.402911124365982, −4.242982076315653, −3.596856391067493, −2.865621078989559, −2.182655759306213, −1.551359983713646, −1.174681901333927, −0.6705325930158297,
0.6705325930158297, 1.174681901333927, 1.551359983713646, 2.182655759306213, 2.865621078989559, 3.596856391067493, 4.242982076315653, 4.402911124365982, 5.217103447834972, 5.509376585172712, 5.966402964652255, 6.545591268754347, 6.940955647917305, 7.474760676397385, 8.162364890010415, 8.513557264974927, 8.746070213711872, 9.435783113212823, 9.941624020861946, 10.58968900879627, 10.77508112618093, 11.21482109690835, 11.70852996612901, 11.92334840769550, 12.70860690386981
