| L(s) = 1 | − 1.41·2-s + 5.76·3-s − 6.00·4-s + 4.58·5-s − 8.15·6-s − 36.4·7-s + 19.7·8-s + 6.23·9-s − 6.48·10-s − 11·11-s − 34.5·12-s + 51.4·14-s + 26.4·15-s + 20.0·16-s − 66.7·17-s − 8.81·18-s − 19.1·19-s − 27.5·20-s − 209.·21-s + 15.5·22-s − 147.·23-s + 114.·24-s − 103.·25-s − 119.·27-s + 218.·28-s − 17.9·29-s − 37.3·30-s + ⋯ |
| L(s) = 1 | − 0.499·2-s + 1.10·3-s − 0.750·4-s + 0.410·5-s − 0.554·6-s − 1.96·7-s + 0.874·8-s + 0.230·9-s − 0.205·10-s − 0.301·11-s − 0.832·12-s + 0.983·14-s + 0.455·15-s + 0.312·16-s − 0.951·17-s − 0.115·18-s − 0.231·19-s − 0.307·20-s − 2.18·21-s + 0.150·22-s − 1.34·23-s + 0.970·24-s − 0.831·25-s − 0.853·27-s + 1.47·28-s − 0.114·29-s − 0.227·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7393743513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7393743513\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.41T + 8T^{2} \) |
| 3 | \( 1 - 5.76T + 27T^{2} \) |
| 5 | \( 1 - 4.58T + 125T^{2} \) |
| 7 | \( 1 + 36.4T + 343T^{2} \) |
| 17 | \( 1 + 66.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 17.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 272.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 44.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 31.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 285.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 436.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 868.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 186.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 688.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 786.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 254.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 768.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 126.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.35e3T + 9.12e5T^{2} \) |
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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
−9.079111175834818256863262935031, −8.210581718995466495078449346398, −7.65936142690559841633376425619, −6.43514251897630306426706206831, −5.95858426321428225482065240592, −4.54507643955406681736370700199, −3.71761020157187898987602462890, −2.90581096437902600954911222274, −2.02406527215232532488308256072, −0.38866408687409483617718916760,
0.38866408687409483617718916760, 2.02406527215232532488308256072, 2.90581096437902600954911222274, 3.71761020157187898987602462890, 4.54507643955406681736370700199, 5.95858426321428225482065240592, 6.43514251897630306426706206831, 7.65936142690559841633376425619, 8.210581718995466495078449346398, 9.079111175834818256863262935031
