| L(s) = 1 | − 2·3-s + 4·5-s + 9-s − 2·11-s − 2·13-s − 8·15-s + 5·17-s − 2·19-s + 23-s + 11·25-s + 4·27-s − 2·29-s + 4·33-s + 4·37-s + 4·39-s − 7·41-s + 4·45-s + 4·47-s − 10·51-s − 8·55-s + 4·57-s − 4·59-s − 8·61-s − 8·65-s − 6·67-s − 2·69-s + 4·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 2.06·15-s + 1.21·17-s − 0.458·19-s + 0.208·23-s + 11/5·25-s + 0.769·27-s − 0.371·29-s + 0.696·33-s + 0.657·37-s + 0.640·39-s − 1.09·41-s + 0.596·45-s + 0.583·47-s − 1.40·51-s − 1.07·55-s + 0.529·57-s − 0.520·59-s − 1.02·61-s − 0.992·65-s − 0.733·67-s − 0.240·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707216332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707216332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−12.49540170729817, −12.01420963355642, −11.68357136865763, −10.96098749231964, −10.62148456891795, −10.25298004151640, −9.975840918792499, −9.390006567329922, −9.045420819500401, −8.447166247493516, −7.835055226152853, −7.320511769960893, −6.800361292324211, −6.268921461655018, −5.923208007256622, −5.573179717891243, −5.124651559079560, −4.809391582896953, −4.162645768397478, −3.192828669588753, −2.849421891639727, −2.240383620537306, −1.625381548834921, −1.134148072250947, −0.3741684844992110,
0.3741684844992110, 1.134148072250947, 1.625381548834921, 2.240383620537306, 2.849421891639727, 3.192828669588753, 4.162645768397478, 4.809391582896953, 5.124651559079560, 5.573179717891243, 5.923208007256622, 6.268921461655018, 6.800361292324211, 7.320511769960893, 7.835055226152853, 8.447166247493516, 9.045420819500401, 9.390006567329922, 9.975840918792499, 10.25298004151640, 10.62148456891795, 10.96098749231964, 11.68357136865763, 12.01420963355642, 12.49540170729817
