L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 3.60·7-s − 2·10-s + 3·11-s − 7.21·14-s − 4·16-s − 3.60·17-s + 7.21·19-s − 2·20-s + 6·22-s + 3.60·23-s + 25-s − 7.21·28-s − 7.21·29-s − 7.21·31-s − 8·32-s − 7.21·34-s + 3.60·35-s − 3.60·37-s + 14.4·38-s + 11·41-s + 4·43-s + 6·44-s + 7.21·46-s − 4·47-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 1.36·7-s − 0.632·10-s + 0.904·11-s − 1.92·14-s − 16-s − 0.874·17-s + 1.65·19-s − 0.447·20-s + 1.27·22-s + 0.751·23-s + 0.200·25-s − 1.36·28-s − 1.33·29-s − 1.29·31-s − 1.41·32-s − 1.23·34-s + 0.609·35-s − 0.592·37-s + 2.33·38-s + 1.71·41-s + 0.609·43-s + 0.904·44-s + 1.06·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.020360378\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.020360378\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2T + 2T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 + 7.21T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 - 11T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - 3.60T + 97T^{2} \) |
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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
−7.40338920510746480309800476729, −7.02107072600789627268707952697, −6.40569271404032458512873440630, −5.61817350694683646713423299377, −5.15524886604877096047680788741, −4.00047718155192463526339852715, −3.75909133339107555969609520747, −3.07512886695855041335256562572, −2.19627800604815603375128958152, −0.67793530494639969641999660350,
0.67793530494639969641999660350, 2.19627800604815603375128958152, 3.07512886695855041335256562572, 3.75909133339107555969609520747, 4.00047718155192463526339852715, 5.15524886604877096047680788741, 5.61817350694683646713423299377, 6.40569271404032458512873440630, 7.02107072600789627268707952697, 7.40338920510746480309800476729