| L(s) = 1 | + 1.41·3-s − 3.16·7-s − 0.999·9-s − 4.47·21-s + 9.48·23-s − 5.65·27-s + 8.94·29-s − 12·41-s − 12.7·43-s + 9.48·47-s + 3.00·49-s − 13.4·61-s + 3.16·63-s − 4.24·67-s + 13.4·69-s − 5.00·81-s + 15.5·83-s + 12.6·87-s − 6·89-s − 8.94·101-s − 15.8·103-s − 18.3·107-s − 13.4·109-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 1.19·7-s − 0.333·9-s − 0.975·21-s + 1.97·23-s − 1.08·27-s + 1.66·29-s − 1.87·41-s − 1.94·43-s + 1.38·47-s + 0.428·49-s − 1.71·61-s + 0.398·63-s − 0.518·67-s + 1.61·69-s − 0.555·81-s + 1.70·83-s + 1.35·87-s − 0.635·89-s − 0.889·101-s − 1.55·103-s − 1.77·107-s − 1.28·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 9.48T + 23T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.79069491138290751032432303008, −6.73554705775518581549603411285, −6.61798698584754146469963017408, −5.49710778668984969280245195189, −4.81429101847871817367810895457, −3.73439817010018474546262549408, −3.04407323497627136313803048011, −2.68968619472845561332936235734, −1.36242191314006362975661551147, 0,
1.36242191314006362975661551147, 2.68968619472845561332936235734, 3.04407323497627136313803048011, 3.73439817010018474546262549408, 4.81429101847871817367810895457, 5.49710778668984969280245195189, 6.61798698584754146469963017408, 6.73554705775518581549603411285, 7.79069491138290751032432303008
