| L(s) = 1 | − 0.267·3-s + 3.78·5-s − 2.20·7-s − 2.92·9-s − 1.08·13-s − 1.01·15-s + 2.61·17-s + 19-s + 0.590·21-s + 1.70·23-s + 9.29·25-s + 1.58·27-s − 7.08·29-s − 2.86·31-s − 8.34·35-s + 3.73·37-s + 0.288·39-s − 2.75·41-s − 6.77·43-s − 11.0·45-s − 2.53·47-s − 2.12·49-s − 0.699·51-s − 10.7·53-s − 0.267·57-s + 8.28·59-s + 7.41·61-s + ⋯ |
| L(s) = 1 | − 0.154·3-s + 1.69·5-s − 0.834·7-s − 0.976·9-s − 0.299·13-s − 0.261·15-s + 0.634·17-s + 0.229·19-s + 0.128·21-s + 0.355·23-s + 1.85·25-s + 0.305·27-s − 1.31·29-s − 0.513·31-s − 1.41·35-s + 0.613·37-s + 0.0462·39-s − 0.429·41-s − 1.03·43-s − 1.65·45-s − 0.370·47-s − 0.303·49-s − 0.0979·51-s − 1.47·53-s − 0.0354·57-s + 1.07·59-s + 0.948·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.267T + 3T^{2} \) |
| 5 | \( 1 - 3.78T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 13 | \( 1 + 1.08T + 13T^{2} \) |
| 17 | \( 1 - 2.61T + 17T^{2} \) |
| 23 | \( 1 - 1.70T + 23T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 + 2.86T + 31T^{2} \) |
| 37 | \( 1 - 3.73T + 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 + 6.77T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 8.28T + 59T^{2} \) |
| 61 | \( 1 - 7.41T + 61T^{2} \) |
| 67 | \( 1 - 5.90T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 4.39T + 79T^{2} \) |
| 83 | \( 1 + 1.18T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 2.27T + 97T^{2} \) |
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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
−7.22222174920006238057842352651, −6.48753592683353102483951624675, −6.00480418315940625887352004864, −5.42301526741929509258311939378, −4.94455843908727321146393194700, −3.59833160758129100953332197523, −2.96708285767007092628856196601, −2.23067981651754462382448557391, −1.34394005581505495827511854698, 0,
1.34394005581505495827511854698, 2.23067981651754462382448557391, 2.96708285767007092628856196601, 3.59833160758129100953332197523, 4.94455843908727321146393194700, 5.42301526741929509258311939378, 6.00480418315940625887352004864, 6.48753592683353102483951624675, 7.22222174920006238057842352651
