| L(s) = 1 | − 2.46·5-s − 3·9-s − 7.92·17-s + 1.07·25-s + 8.46·29-s − 9.39·37-s + 1.92·41-s + 7.39·45-s − 7·49-s − 10.4·53-s − 5.39·61-s + 16.8·73-s + 9·81-s + 19.5·85-s + 10·89-s + 18·97-s − 16.3·101-s + 6·109-s + 20.8·113-s + ⋯ |
| L(s) = 1 | − 1.10·5-s − 9-s − 1.92·17-s + 0.214·25-s + 1.57·29-s − 1.54·37-s + 0.301·41-s + 1.10·45-s − 49-s − 1.43·53-s − 0.690·61-s + 1.97·73-s + 81-s + 2.11·85-s + 1.05·89-s + 1.82·97-s − 1.62·101-s + 0.574·109-s + 1.96·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7055291463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7055291463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8.46T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.39T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−8.259940276924542474205691752400, −7.56251640150692743131899460682, −6.65470608501995748967457375573, −6.24545166216826244940788587369, −5.06024215683950327731809265843, −4.55691729694495304277263072565, −3.66054072176450290699381826535, −2.93410897743996099493501531590, −1.98012986105323017224554684664, −0.42824352894671353067376791343,
0.42824352894671353067376791343, 1.98012986105323017224554684664, 2.93410897743996099493501531590, 3.66054072176450290699381826535, 4.55691729694495304277263072565, 5.06024215683950327731809265843, 6.24545166216826244940788587369, 6.65470608501995748967457375573, 7.56251640150692743131899460682, 8.259940276924542474205691752400
