| 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
| \(L(1)\) |
\(\approx\) |
\(1.080721602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080721602\) |
| \(L(\frac{3}{2})\) |
|
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
−8.862133375291952360222808257523, −8.313550274550052274072416963525, −7.71961054666166689607552563894, −6.68238811525108223837190753596, −6.10320096753296583924939336118, −4.90640372836009974697442855441, −4.12566059157906538560690873193, −3.10559347532886370234082178054, −1.68191823746829392101927104267, −0.848118109209223973030578593831,
0.848118109209223973030578593831, 1.68191823746829392101927104267, 3.10559347532886370234082178054, 4.12566059157906538560690873193, 4.90640372836009974697442855441, 6.10320096753296583924939336118, 6.68238811525108223837190753596, 7.71961054666166689607552563894, 8.313550274550052274072416963525, 8.862133375291952360222808257523
