L(s) = 1 | − 2·4-s − 3.60·5-s − 3·9-s + 4·16-s − 3.60·19-s + 7.21·20-s − 23-s + 7.99·25-s − 5·29-s + 10.8·31-s + 6·36-s − 7.21·41-s − 9·43-s + 10.8·45-s + 3.60·47-s + 11·53-s + 14.4·59-s − 8·64-s − 10.8·73-s + 7.21·76-s + 15·79-s − 14.4·80-s + 9·81-s + 18.0·83-s + 3.60·89-s + 2·92-s + 12.9·95-s + ⋯ |
L(s) = 1 | − 4-s − 1.61·5-s − 9-s + 16-s − 0.827·19-s + 1.61·20-s − 0.208·23-s + 1.59·25-s − 0.928·29-s + 1.94·31-s + 36-s − 1.12·41-s − 1.37·43-s + 1.61·45-s + 0.525·47-s + 1.51·53-s + 1.87·59-s − 64-s − 1.26·73-s + 0.827·76-s + 1.68·79-s − 1.61·80-s + 81-s + 1.97·83-s + 0.382·89-s + 0.208·92-s + 1.33·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 7.21T + 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 - 3.60T + 47T^{2} \) |
| 53 | \( 1 - 11T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 - 3.60T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−7.69913170685552903296299653110, −6.84881867125274396872993919539, −6.05860702388404523739010663704, −5.14653537581162888608739552742, −4.63096275524713101713774358868, −3.78647866064763939135712329726, −3.44662570850893324070288468305, −2.37350425892920205370682200506, −0.800506457038119216787322834861, 0,
0.800506457038119216787322834861, 2.37350425892920205370682200506, 3.44662570850893324070288468305, 3.78647866064763939135712329726, 4.63096275524713101713774358868, 5.14653537581162888608739552742, 6.05860702388404523739010663704, 6.84881867125274396872993919539, 7.69913170685552903296299653110