| L(s) = 1 | + 0.792·3-s − 0.983·5-s + 1.99·7-s − 2.37·9-s + 1.73·11-s + 3.28·13-s − 0.779·15-s + 17-s − 2.40·19-s + 1.58·21-s + 7.43·23-s − 4.03·25-s − 4.25·27-s + 1.18·29-s − 2.58·31-s + 1.37·33-s − 1.96·35-s + 5.78·37-s + 2.59·39-s + 5.60·41-s − 5.69·43-s + 2.33·45-s + 8.65·47-s − 3.00·49-s + 0.792·51-s − 3.71·53-s − 1.70·55-s + ⋯ |
| L(s) = 1 | + 0.457·3-s − 0.439·5-s + 0.755·7-s − 0.790·9-s + 0.523·11-s + 0.909·13-s − 0.201·15-s + 0.242·17-s − 0.551·19-s + 0.345·21-s + 1.55·23-s − 0.806·25-s − 0.818·27-s + 0.219·29-s − 0.464·31-s + 0.239·33-s − 0.332·35-s + 0.951·37-s + 0.416·39-s + 0.874·41-s − 0.869·43-s + 0.347·45-s + 1.26·47-s − 0.429·49-s + 0.110·51-s − 0.510·53-s − 0.230·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.463032964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.463032964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 0.792T + 3T^{2} \) |
| 5 | \( 1 + 0.983T + 5T^{2} \) |
| 7 | \( 1 - 1.99T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 19 | \( 1 + 2.40T + 19T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 - 5.78T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 - 8.65T + 47T^{2} \) |
| 53 | \( 1 + 3.71T + 53T^{2} \) |
| 61 | \( 1 - 3.84T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 3.48T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−8.013663269305536828410646426011, −7.27918841827679022148974394714, −6.44018028373993100967945541515, −5.76752848951753306337861285903, −5.00709763658765276461677646656, −4.15549789912861285354119400214, −3.54694566930563046949612374104, −2.73339561186908647991278106122, −1.78737629597883197264452618843, −0.77860262435401815349207058534,
0.77860262435401815349207058534, 1.78737629597883197264452618843, 2.73339561186908647991278106122, 3.54694566930563046949612374104, 4.15549789912861285354119400214, 5.00709763658765276461677646656, 5.76752848951753306337861285903, 6.44018028373993100967945541515, 7.27918841827679022148974394714, 8.013663269305536828410646426011
