| L(s) = 1 | − 2.23·3-s − 2·4-s + 2.00·9-s − 3·11-s + 4.47·12-s + 6.70·13-s + 4·16-s + 2.23·17-s + 2.23·27-s − 9·29-s + 6.70·33-s − 4.00·36-s − 15.0·39-s + 6·44-s − 11.1·47-s − 8.94·48-s − 5.00·51-s − 13.4·52-s − 8·64-s − 4.47·68-s − 12·71-s + 13.4·73-s − 79-s − 11·81-s − 8.94·83-s + 20.1·87-s − 6.70·97-s + ⋯ |
| L(s) = 1 | − 1.29·3-s − 4-s + 0.666·9-s − 0.904·11-s + 1.29·12-s + 1.86·13-s + 16-s + 0.542·17-s + 0.430·27-s − 1.67·29-s + 1.16·33-s − 0.666·36-s − 2.40·39-s + 0.904·44-s − 1.63·47-s − 1.29·48-s − 0.700·51-s − 1.86·52-s − 64-s − 0.542·68-s − 1.42·71-s + 1.57·73-s − 0.112·79-s − 1.22·81-s − 0.981·83-s + 2.15·87-s − 0.681·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−9.388184580540948124959442289420, −8.479529323929461630967714085177, −7.80297977430087953436193082919, −6.56743779994051427602049302435, −5.67600780303091926205299634625, −5.32168678165937101325745264997, −4.23446721686868124359633520296, −3.30678461771693015492849077192, −1.30915866113986890771182369008, 0,
1.30915866113986890771182369008, 3.30678461771693015492849077192, 4.23446721686868124359633520296, 5.32168678165937101325745264997, 5.67600780303091926205299634625, 6.56743779994051427602049302435, 7.80297977430087953436193082919, 8.479529323929461630967714085177, 9.388184580540948124959442289420
