L(s) = 1 | − 2.60·2-s + 3.18·3-s + 4.76·4-s − 0.824·5-s − 8.29·6-s − 1.42·7-s − 7.19·8-s + 7.16·9-s + 2.14·10-s + 1.79·11-s + 15.1·12-s + 2.40·13-s + 3.70·14-s − 2.62·15-s + 9.19·16-s + 4.37·17-s − 18.6·18-s + 4.55·19-s − 3.92·20-s − 4.54·21-s − 4.67·22-s − 7.24·23-s − 22.9·24-s − 4.32·25-s − 6.26·26-s + 13.2·27-s − 6.79·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s + 1.84·3-s + 2.38·4-s − 0.368·5-s − 3.38·6-s − 0.538·7-s − 2.54·8-s + 2.38·9-s + 0.678·10-s + 0.542·11-s + 4.38·12-s + 0.667·13-s + 0.991·14-s − 0.678·15-s + 2.29·16-s + 1.06·17-s − 4.39·18-s + 1.04·19-s − 0.878·20-s − 0.992·21-s − 0.997·22-s − 1.50·23-s − 4.68·24-s − 0.864·25-s − 1.22·26-s + 2.55·27-s − 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.714050254\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.714050254\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5077 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 - 3.18T + 3T^{2} \) |
| 5 | \( 1 + 0.824T + 5T^{2} \) |
| 7 | \( 1 + 1.42T + 7T^{2} \) |
| 11 | \( 1 - 1.79T + 11T^{2} \) |
| 13 | \( 1 - 2.40T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 - 0.631T + 29T^{2} \) |
| 31 | \( 1 + 1.35T + 31T^{2} \) |
| 37 | \( 1 + 6.27T + 37T^{2} \) |
| 41 | \( 1 - 8.27T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 9.30T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 - 2.29T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.335313977245963534561864118452, −7.81016057457754066016191391216, −7.27264720000065426573713147880, −6.63215303190527020860668202724, −5.62816447028152158546460202351, −3.80304740600901732849979458629, −3.58911801838879358244751648482, −2.55958926414726360724569714900, −1.80016660898889980269393700107, −0.891638088435844625577687381371,
0.891638088435844625577687381371, 1.80016660898889980269393700107, 2.55958926414726360724569714900, 3.58911801838879358244751648482, 3.80304740600901732849979458629, 5.62816447028152158546460202351, 6.63215303190527020860668202724, 7.27264720000065426573713147880, 7.81016057457754066016191391216, 8.335313977245963534561864118452