| L(s) = 1 | − 2.45·3-s + 4.99·5-s − 13.8·7-s − 20.9·9-s − 39.1·11-s − 68.8·13-s − 12.2·15-s − 73.2·19-s + 34.0·21-s − 110.·23-s − 100.·25-s + 117.·27-s − 70.1·29-s + 13.9·31-s + 96.1·33-s − 69.1·35-s − 201.·37-s + 169.·39-s − 358.·41-s − 548.·43-s − 104.·45-s + 264.·47-s − 151.·49-s + 632.·53-s − 195.·55-s + 180.·57-s + 186.·59-s + ⋯ |
| L(s) = 1 | − 0.472·3-s + 0.446·5-s − 0.747·7-s − 0.776·9-s − 1.07·11-s − 1.46·13-s − 0.211·15-s − 0.885·19-s + 0.353·21-s − 1.00·23-s − 0.800·25-s + 0.840·27-s − 0.449·29-s + 0.0806·31-s + 0.507·33-s − 0.334·35-s − 0.893·37-s + 0.695·39-s − 1.36·41-s − 1.94·43-s − 0.346·45-s + 0.821·47-s − 0.440·49-s + 1.63·53-s − 0.479·55-s + 0.418·57-s + 0.411·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.080102129\times10^{-8}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.080102129\times10^{-8}\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2.45T + 27T^{2} \) |
| 5 | \( 1 - 4.99T + 125T^{2} \) |
| 7 | \( 1 + 13.8T + 343T^{2} \) |
| 11 | \( 1 + 39.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.8T + 2.19e3T^{2} \) |
| 19 | \( 1 + 73.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 70.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 13.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 358.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 548.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 264.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 632.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 186.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 287.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 4.09T + 3.00e5T^{2} \) |
| 71 | \( 1 - 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 454.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 442.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 921.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 104.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 836.T + 9.12e5T^{2} \) |
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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
−8.612531802076307359325013312506, −7.912382578926249787301593833141, −6.97410744233853558272628412298, −6.26918339558637062374891215809, −5.45521370177970728438079856320, −4.96615662999999564329261328437, −3.71432118830704856887809859402, −2.67827314639564065957109702543, −1.99861734004036248013845094125, −0.000084368435479017848774304608,
0.000084368435479017848774304608, 1.99861734004036248013845094125, 2.67827314639564065957109702543, 3.71432118830704856887809859402, 4.96615662999999564329261328437, 5.45521370177970728438079856320, 6.26918339558637062374891215809, 6.97410744233853558272628412298, 7.912382578926249787301593833141, 8.612531802076307359325013312506
