| L(s) = 1 | − 2.75·3-s + 0.520·5-s − 3.84·7-s + 4.59·9-s + 1.15·11-s + 5.55·13-s − 1.43·15-s − 2·17-s + 1.68·19-s + 10.5·21-s + 9.24·23-s − 4.72·25-s − 4.40·27-s − 8.28·29-s − 8.99·31-s − 3.17·33-s − 2·35-s + 37-s − 15.3·39-s − 7.82·41-s + 8.06·43-s + 2.39·45-s − 4.62·47-s + 7.76·49-s + 5.51·51-s + 2.60·53-s + 0.598·55-s + ⋯ |
| L(s) = 1 | − 1.59·3-s + 0.232·5-s − 1.45·7-s + 1.53·9-s + 0.346·11-s + 1.54·13-s − 0.370·15-s − 0.485·17-s + 0.386·19-s + 2.31·21-s + 1.92·23-s − 0.945·25-s − 0.848·27-s − 1.53·29-s − 1.61·31-s − 0.552·33-s − 0.338·35-s + 0.164·37-s − 2.45·39-s − 1.22·41-s + 1.23·43-s + 0.356·45-s − 0.675·47-s + 1.10·49-s + 0.772·51-s + 0.357·53-s + 0.0807·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.75T + 3T^{2} \) |
| 5 | \( 1 - 0.520T + 5T^{2} \) |
| 7 | \( 1 + 3.84T + 7T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 23 | \( 1 - 9.24T + 23T^{2} \) |
| 29 | \( 1 + 8.28T + 29T^{2} \) |
| 31 | \( 1 + 8.99T + 31T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 + 4.62T + 47T^{2} \) |
| 53 | \( 1 - 2.60T + 53T^{2} \) |
| 59 | \( 1 + 3.04T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 + 5.78T + 67T^{2} \) |
| 71 | \( 1 + 1.19T + 71T^{2} \) |
| 73 | \( 1 + 5.42T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 4.89T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−9.426921793105508063998122733739, −8.852626055121826721259235669626, −7.24755727910949681418669675627, −6.70427094418592257487532822618, −5.88091818234362450709100706753, −5.48989573156130063054326031597, −4.12958533399275017902062046545, −3.25874116362296927314404873251, −1.39899528391069048440511480539, 0,
1.39899528391069048440511480539, 3.25874116362296927314404873251, 4.12958533399275017902062046545, 5.48989573156130063054326031597, 5.88091818234362450709100706753, 6.70427094418592257487532822618, 7.24755727910949681418669675627, 8.852626055121826721259235669626, 9.426921793105508063998122733739
