| L(s) = 1 | + 2.35·2-s + 3.56·4-s + 4.18·5-s − 2.82·7-s + 3.69·8-s + 9.86·10-s − 2.48·11-s + 4.60·13-s − 6.67·14-s + 1.59·16-s + 5.70·17-s + 0.542·19-s + 14.9·20-s − 5.86·22-s − 6.30·23-s + 12.4·25-s + 10.8·26-s − 10.0·28-s + 4.16·29-s + 5.03·31-s − 3.64·32-s + 13.4·34-s − 11.8·35-s − 4.02·37-s + 1.28·38-s + 15.4·40-s + 2.04·41-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.78·4-s + 1.87·5-s − 1.06·7-s + 1.30·8-s + 3.12·10-s − 0.749·11-s + 1.27·13-s − 1.78·14-s + 0.397·16-s + 1.38·17-s + 0.124·19-s + 3.33·20-s − 1.25·22-s − 1.31·23-s + 2.49·25-s + 2.12·26-s − 1.90·28-s + 0.774·29-s + 0.905·31-s − 0.644·32-s + 2.30·34-s − 2.00·35-s − 0.661·37-s + 0.207·38-s + 2.44·40-s + 0.319·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.226616595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.226616595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 4.60T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 - 0.542T + 19T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 - 4.16T + 29T^{2} \) |
| 31 | \( 1 - 5.03T + 31T^{2} \) |
| 37 | \( 1 + 4.02T + 37T^{2} \) |
| 41 | \( 1 - 2.04T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 - 8.60T + 53T^{2} \) |
| 59 | \( 1 + 2.15T + 59T^{2} \) |
| 61 | \( 1 + 8.53T + 61T^{2} \) |
| 67 | \( 1 + 3.63T + 67T^{2} \) |
| 71 | \( 1 + 7.43T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
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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
−8.170805278692170952131366724509, −7.07072177526066911859915328775, −6.34295794591794669689377172128, −5.79065465225668054940741870274, −5.68373525521482584363603640432, −4.66781715338169991237066232481, −3.67695954316407786512205230402, −2.95345525387437185225192369148, −2.36347446783534276342643972329, −1.26049235092130611997725984632,
1.26049235092130611997725984632, 2.36347446783534276342643972329, 2.95345525387437185225192369148, 3.67695954316407786512205230402, 4.66781715338169991237066232481, 5.68373525521482584363603640432, 5.79065465225668054940741870274, 6.34295794591794669689377172128, 7.07072177526066911859915328775, 8.170805278692170952131366724509
