| L(s) = 1 | + 1.53·2-s + 0.369·4-s − 0.290·7-s − 2.51·8-s + 11-s − 6.97·13-s − 0.447·14-s − 4.60·16-s + 4.78·17-s + 7.75·19-s + 1.53·22-s + 4·23-s − 10.7·26-s − 0.107·28-s + 7.41·29-s + 6.34·31-s − 2.06·32-s + 7.36·34-s + 3.41·37-s + 11.9·38-s + 7.41·41-s − 0.290·43-s + 0.369·44-s + 6.15·46-s + 5.26·47-s − 6.91·49-s − 2.57·52-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.184·4-s − 0.109·7-s − 0.887·8-s + 0.301·11-s − 1.93·13-s − 0.119·14-s − 1.15·16-s + 1.16·17-s + 1.77·19-s + 0.328·22-s + 0.834·23-s − 2.10·26-s − 0.0202·28-s + 1.37·29-s + 1.13·31-s − 0.364·32-s + 1.26·34-s + 0.562·37-s + 1.93·38-s + 1.15·41-s − 0.0443·43-s + 0.0556·44-s + 0.907·46-s + 0.767·47-s − 0.987·49-s − 0.356·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.729027159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.729027159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 7 | \( 1 + 0.290T + 7T^{2} \) |
| 13 | \( 1 + 6.97T + 13T^{2} \) |
| 17 | \( 1 - 4.78T + 17T^{2} \) |
| 19 | \( 1 - 7.75T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 7.41T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 + 0.290T + 43T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 - 5.75T + 53T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 6.15T + 67T^{2} \) |
| 71 | \( 1 - 5.07T + 71T^{2} \) |
| 73 | \( 1 - 1.12T + 73T^{2} \) |
| 79 | \( 1 + 0.921T + 79T^{2} \) |
| 83 | \( 1 - 1.70T + 83T^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−9.137979202138872531961689909987, −7.949502648462026233065355822174, −7.31913803183067662321325352841, −6.46611353932453640354512461667, −5.52801978689900725203399471533, −4.97347032591387060146462746008, −4.30340026516154545582161967472, −3.10488219441085891518929142840, −2.71344478438705347354592336717, −0.910366465465038581550271385489,
0.910366465465038581550271385489, 2.71344478438705347354592336717, 3.10488219441085891518929142840, 4.30340026516154545582161967472, 4.97347032591387060146462746008, 5.52801978689900725203399471533, 6.46611353932453640354512461667, 7.31913803183067662321325352841, 7.949502648462026233065355822174, 9.137979202138872531961689909987
