| L(s) = 1 | − 7-s + 11-s − 13-s − 6·17-s − 7·19-s − 6·23-s − 6·29-s + 7·31-s + 2·37-s + 6·41-s + 43-s − 6·49-s + 6·53-s − 5·61-s − 5·67-s − 12·71-s − 14·73-s − 77-s + 4·79-s − 6·83-s − 6·89-s + 91-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s − 0.277·13-s − 1.45·17-s − 1.60·19-s − 1.25·23-s − 1.11·29-s + 1.25·31-s + 0.328·37-s + 0.937·41-s + 0.152·43-s − 6/7·49-s + 0.824·53-s − 0.640·61-s − 0.610·67-s − 1.42·71-s − 1.63·73-s − 0.113·77-s + 0.450·79-s − 0.658·83-s − 0.635·89-s + 0.104·91-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.61345949348824, −13.25699580530457, −12.89059865964133, −12.39236841801841, −11.77881790757318, −11.51213742595155, −10.77978779189295, −10.52550623277378, −9.976686838865119, −9.303503114589420, −9.125722920208120, −8.339837905140474, −8.167360223315966, −7.355647271352543, −6.934602968136403, −6.281293082662687, −6.110983229352172, −5.467705569610782, −4.574495515233263, −4.249122198232068, −3.953318711961968, −2.982512918897517, −2.487488039294198, −1.959833718039766, −1.272195192836488, 0, 0,
1.272195192836488, 1.959833718039766, 2.487488039294198, 2.982512918897517, 3.953318711961968, 4.249122198232068, 4.574495515233263, 5.467705569610782, 6.110983229352172, 6.281293082662687, 6.934602968136403, 7.355647271352543, 8.167360223315966, 8.339837905140474, 9.125722920208120, 9.303503114589420, 9.976686838865119, 10.52550623277378, 10.77978779189295, 11.51213742595155, 11.77881790757318, 12.39236841801841, 12.89059865964133, 13.25699580530457, 13.61345949348824
