| L(s) = 1 | + 5-s + 13-s + 6·17-s + 4·19-s + 25-s + 2·29-s − 2·37-s + 2·41-s − 4·43-s − 4·47-s − 7·49-s + 10·53-s − 8·59-s − 2·61-s + 65-s − 4·67-s + 12·71-s − 6·73-s + 16·83-s + 6·85-s + 10·89-s + 4·95-s + 2·97-s + 10·101-s − 8·103-s − 12·107-s + 6·109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.277·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s − 0.328·37-s + 0.312·41-s − 0.609·43-s − 0.583·47-s − 49-s + 1.37·53-s − 1.04·59-s − 0.256·61-s + 0.124·65-s − 0.488·67-s + 1.42·71-s − 0.702·73-s + 1.75·83-s + 0.650·85-s + 1.05·89-s + 0.410·95-s + 0.203·97-s + 0.995·101-s − 0.788·103-s − 1.16·107-s + 0.574·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.354673197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.354673197\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.205428436133657280472376869439, −7.65900647239731985631264028860, −6.84309453834556716180176059884, −6.06228021687514875709261074855, −5.40267935401785701535254251049, −4.73371770359721681749687859177, −3.58625364905200255174697777626, −3.01240117704288435947790477909, −1.83776004791737658204696996390, −0.893179867382728641044098248034,
0.893179867382728641044098248034, 1.83776004791737658204696996390, 3.01240117704288435947790477909, 3.58625364905200255174697777626, 4.73371770359721681749687859177, 5.40267935401785701535254251049, 6.06228021687514875709261074855, 6.84309453834556716180176059884, 7.65900647239731985631264028860, 8.205428436133657280472376869439
