| L(s) = 1 | − 2.54·2-s − 1.60·3-s + 4.48·4-s − 2.62·5-s + 4.09·6-s + 1.52·7-s − 6.32·8-s − 0.419·9-s + 6.69·10-s + 5.10·11-s − 7.20·12-s − 1.33·13-s − 3.88·14-s + 4.22·15-s + 7.13·16-s + 8.04·17-s + 1.06·18-s + 1.96·19-s − 11.7·20-s − 2.45·21-s − 13.0·22-s − 23-s + 10.1·24-s + 1.90·25-s + 3.40·26-s + 5.49·27-s + 6.84·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.927·3-s + 2.24·4-s − 1.17·5-s + 1.66·6-s + 0.577·7-s − 2.23·8-s − 0.139·9-s + 2.11·10-s + 1.54·11-s − 2.07·12-s − 0.371·13-s − 1.03·14-s + 1.08·15-s + 1.78·16-s + 1.95·17-s + 0.251·18-s + 0.451·19-s − 2.63·20-s − 0.535·21-s − 2.77·22-s − 0.208·23-s + 2.07·24-s + 0.381·25-s + 0.668·26-s + 1.05·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 + T \) |
| 349 | \( 1 + T \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 + 1.60T + 3T^{2} \) |
| 5 | \( 1 + 2.62T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 5.10T + 11T^{2} \) |
| 13 | \( 1 + 1.33T + 13T^{2} \) |
| 17 | \( 1 - 8.04T + 17T^{2} \) |
| 19 | \( 1 - 1.96T + 19T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 + 0.387T + 31T^{2} \) |
| 37 | \( 1 + 2.04T + 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 - 3.28T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 6.27T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 1.99T + 61T^{2} \) |
| 67 | \( 1 + 5.26T + 67T^{2} \) |
| 71 | \( 1 + 1.29T + 71T^{2} \) |
| 73 | \( 1 + 7.26T + 73T^{2} \) |
| 79 | \( 1 + 0.403T + 79T^{2} \) |
| 83 | \( 1 + 0.356T + 83T^{2} \) |
| 89 | \( 1 + 9.46T + 89T^{2} \) |
| 97 | \( 1 + 4.67T + 97T^{2} \) |
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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
−7.59616903797603916258215681998, −7.12076956644393128979070289822, −6.33377539976076292922473528705, −5.67931497600539759534084125429, −4.75931311711637801907369782338, −3.75337354703821447909213683329, −2.98009485733088408367421733085, −1.57271560074096532963507631010, −0.983456683533313378247511928952, 0,
0.983456683533313378247511928952, 1.57271560074096532963507631010, 2.98009485733088408367421733085, 3.75337354703821447909213683329, 4.75931311711637801907369782338, 5.67931497600539759534084125429, 6.33377539976076292922473528705, 7.12076956644393128979070289822, 7.59616903797603916258215681998
