L(s) = 1 | + 1.30·2-s + 3.30·3-s − 0.302·4-s + 3·5-s + 4.30·6-s − 3·8-s + 7.90·9-s + 3.90·10-s − 4.30·11-s − 1.00·12-s − 1.60·13-s + 9.90·15-s − 3.30·16-s + 1.69·17-s + 10.3·18-s − 19-s − 0.908·20-s − 5.60·22-s − 3·23-s − 9.90·24-s + 4·25-s − 2.09·26-s + 16.2·27-s − 0.908·29-s + 12.9·30-s + 2.30·31-s + 1.69·32-s + ⋯ |
L(s) = 1 | + 0.921·2-s + 1.90·3-s − 0.151·4-s + 1.34·5-s + 1.75·6-s − 1.06·8-s + 2.63·9-s + 1.23·10-s − 1.29·11-s − 0.288·12-s − 0.445·13-s + 2.55·15-s − 0.825·16-s + 0.411·17-s + 2.42·18-s − 0.229·19-s − 0.203·20-s − 1.19·22-s − 0.625·23-s − 2.02·24-s + 0.800·25-s − 0.410·26-s + 3.11·27-s − 0.168·29-s + 2.35·30-s + 0.413·31-s + 0.300·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.643618647\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.643618647\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 - 1.69T + 17T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 0.908T + 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 + 4.30T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 - 8.21T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 8.90T + 67T^{2} \) |
| 71 | \( 1 + 2.21T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 3.21T + 79T^{2} \) |
| 83 | \( 1 - 2.09T + 83T^{2} \) |
| 89 | \( 1 - 3.39T + 89T^{2} \) |
| 97 | \( 1 + 2.39T + 97T^{2} \) |
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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
−9.909690724418764979675476989283, −9.251484283982329155265377174552, −8.465157922028266551654679290940, −7.71628845206791363792837877103, −6.58684002998486052032557530845, −5.47174431640610847000982248692, −4.68743584837260494509493933048, −3.53785132501114809093491309714, −2.70717446011079999376010246326, −1.96632893080951077507603908848,
1.96632893080951077507603908848, 2.70717446011079999376010246326, 3.53785132501114809093491309714, 4.68743584837260494509493933048, 5.47174431640610847000982248692, 6.58684002998486052032557530845, 7.71628845206791363792837877103, 8.465157922028266551654679290940, 9.251484283982329155265377174552, 9.909690724418764979675476989283