| L(s) = 1 | − 0.836·3-s − 2.30·9-s + 0.403·11-s − 6.20·13-s − 2.62·17-s + 6.41·19-s − 6.54·23-s + 4.43·27-s + 1.96·29-s + 4.66·31-s − 0.337·33-s + 3.44·37-s + 5.19·39-s + 2.68·41-s + 10.8·43-s + 9.67·47-s + 2.19·51-s + 5.97·53-s − 5.36·57-s + 9.18·59-s + 5.69·61-s − 11.8·67-s + 5.47·69-s − 0.530·71-s − 8.20·73-s − 9.12·79-s + 3.19·81-s + ⋯ |
| L(s) = 1 | − 0.483·3-s − 0.766·9-s + 0.121·11-s − 1.72·13-s − 0.636·17-s + 1.47·19-s − 1.36·23-s + 0.853·27-s + 0.364·29-s + 0.838·31-s − 0.0587·33-s + 0.566·37-s + 0.831·39-s + 0.418·41-s + 1.65·43-s + 1.41·47-s + 0.307·51-s + 0.820·53-s − 0.710·57-s + 1.19·59-s + 0.729·61-s − 1.45·67-s + 0.659·69-s − 0.0629·71-s − 0.959·73-s − 1.02·79-s + 0.354·81-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 | \( 1 + 0.836T + 3T^{2} \) |
| 11 | \( 1 - 0.403T + 11T^{2} \) |
| 13 | \( 1 + 6.20T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 + 6.54T + 23T^{2} \) |
| 29 | \( 1 - 1.96T + 29T^{2} \) |
| 31 | \( 1 - 4.66T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 9.67T + 47T^{2} \) |
| 53 | \( 1 - 5.97T + 53T^{2} \) |
| 59 | \( 1 - 9.18T + 59T^{2} \) |
| 61 | \( 1 - 5.69T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 0.530T + 71T^{2} \) |
| 73 | \( 1 + 8.20T + 73T^{2} \) |
| 79 | \( 1 + 9.12T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.41152016866590775950483038365, −6.63547200118074729391020447057, −5.81684294852377251961464499295, −5.42110393875823333515014261873, −4.59188087629241471678793335877, −3.97746652667970689059093206285, −2.70183030631651997503311789817, −2.47178946412004219993377309239, −1.01850827907077423068255242001, 0,
1.01850827907077423068255242001, 2.47178946412004219993377309239, 2.70183030631651997503311789817, 3.97746652667970689059093206285, 4.59188087629241471678793335877, 5.42110393875823333515014261873, 5.81684294852377251961464499295, 6.63547200118074729391020447057, 7.41152016866590775950483038365
