| L(s) = 1 | + 3-s − 2·9-s + 3·11-s − 13-s + 5·17-s − 6·19-s − 5·27-s − 5·29-s + 2·31-s + 3·33-s + 4·37-s − 39-s − 2·41-s − 10·43-s + 9·47-s + 5·51-s − 6·53-s − 6·57-s − 6·59-s − 12·61-s + 2·67-s + 14·73-s + 79-s + 81-s − 12·83-s − 5·87-s + 2·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.904·11-s − 0.277·13-s + 1.21·17-s − 1.37·19-s − 0.962·27-s − 0.928·29-s + 0.359·31-s + 0.522·33-s + 0.657·37-s − 0.160·39-s − 0.312·41-s − 1.52·43-s + 1.31·47-s + 0.700·51-s − 0.824·53-s − 0.794·57-s − 0.781·59-s − 1.53·61-s + 0.244·67-s + 1.63·73-s + 0.112·79-s + 1/9·81-s − 1.31·83-s − 0.536·87-s + 0.207·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−7.44965394339439066417022514371, −6.60773924481684227360224826190, −6.03038797854653195128850773564, −5.33254898584901439823949986516, −4.43014986611520148003595350681, −3.71294788300980600019747172276, −3.06599606111858182482227650526, −2.21791907869773500408663138909, −1.35577972239864795625856007293, 0,
1.35577972239864795625856007293, 2.21791907869773500408663138909, 3.06599606111858182482227650526, 3.71294788300980600019747172276, 4.43014986611520148003595350681, 5.33254898584901439823949986516, 6.03038797854653195128850773564, 6.60773924481684227360224826190, 7.44965394339439066417022514371
