| L(s) = 1 | + 1.25·2-s + 3-s − 0.412·4-s − 1.76·5-s + 1.25·6-s − 1.24·7-s − 3.03·8-s + 9-s − 2.22·10-s + 5.12·11-s − 0.412·12-s − 13-s − 1.56·14-s − 1.76·15-s − 3.00·16-s + 0.471·17-s + 1.25·18-s + 1.33·19-s + 0.729·20-s − 1.24·21-s + 6.45·22-s − 1.30·23-s − 3.03·24-s − 1.87·25-s − 1.25·26-s + 27-s + 0.513·28-s + ⋯ |
| L(s) = 1 | + 0.890·2-s + 0.577·3-s − 0.206·4-s − 0.789·5-s + 0.514·6-s − 0.470·7-s − 1.07·8-s + 0.333·9-s − 0.703·10-s + 1.54·11-s − 0.119·12-s − 0.277·13-s − 0.419·14-s − 0.456·15-s − 0.751·16-s + 0.114·17-s + 0.296·18-s + 0.305·19-s + 0.163·20-s − 0.271·21-s + 1.37·22-s − 0.271·23-s − 0.620·24-s − 0.375·25-s − 0.247·26-s + 0.192·27-s + 0.0970·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.25T + 2T^{2} \) |
| 5 | \( 1 + 1.76T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 17 | \( 1 - 0.471T + 17T^{2} \) |
| 19 | \( 1 - 1.33T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 1.98T + 29T^{2} \) |
| 31 | \( 1 + 3.93T + 31T^{2} \) |
| 37 | \( 1 - 6.27T + 37T^{2} \) |
| 41 | \( 1 - 0.533T + 41T^{2} \) |
| 43 | \( 1 + 3.75T + 43T^{2} \) |
| 47 | \( 1 + 4.27T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 8.69T + 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 71 | \( 1 + 9.68T + 71T^{2} \) |
| 73 | \( 1 + 7.25T + 73T^{2} \) |
| 79 | \( 1 + 6.17T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 0.725T + 97T^{2} \) |
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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
−8.062771351408504640923771510747, −7.33062095697794503276887662146, −6.47536861869506172221089724745, −5.88810266502806466017428169346, −4.77055195636165302154970352309, −4.15433734831224016558888000528, −3.54911099489186892239097174404, −2.96109529355146561933618384121, −1.55454238515421423939922920450, 0,
1.55454238515421423939922920450, 2.96109529355146561933618384121, 3.54911099489186892239097174404, 4.15433734831224016558888000528, 4.77055195636165302154970352309, 5.88810266502806466017428169346, 6.47536861869506172221089724745, 7.33062095697794503276887662146, 8.062771351408504640923771510747
