| L(s) = 1 | − 3-s − 3.00·5-s + 4.42·7-s + 9-s − 1.26·11-s − 7.02·13-s + 3.00·15-s + 4.46·17-s − 6.47·19-s − 4.42·21-s − 5.72·23-s + 4.04·25-s − 27-s − 5.47·29-s + 3.83·31-s + 1.26·33-s − 13.3·35-s + 1.66·37-s + 7.02·39-s + 3.44·41-s + 7.06·43-s − 3.00·45-s + 2.66·47-s + 12.5·49-s − 4.46·51-s − 11.5·53-s + 3.79·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.67·7-s + 0.333·9-s − 0.380·11-s − 1.94·13-s + 0.776·15-s + 1.08·17-s − 1.48·19-s − 0.965·21-s − 1.19·23-s + 0.808·25-s − 0.192·27-s − 1.01·29-s + 0.687·31-s + 0.219·33-s − 2.24·35-s + 0.274·37-s + 1.12·39-s + 0.538·41-s + 1.07·43-s − 0.448·45-s + 0.388·47-s + 1.79·49-s − 0.625·51-s − 1.58·53-s + 0.511·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8508215625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8508215625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 3.00T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 7.02T + 13T^{2} \) |
| 17 | \( 1 - 4.46T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 + 5.47T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 - 3.44T + 41T^{2} \) |
| 43 | \( 1 - 7.06T + 43T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 8.23T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 7.21T + 67T^{2} \) |
| 71 | \( 1 - 3.73T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 5.90T + 79T^{2} \) |
| 83 | \( 1 - 2.99T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 9.56T + 97T^{2} \) |
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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
−7.974925935330775927725964787199, −7.55404977772972906046212316126, −6.90293156584032551939407351109, −5.69254582856676851595706442169, −5.16793898148002160130131372872, −4.27925277160459172480525317097, −4.14744656494423976745085298384, −2.65862304203081840648378457415, −1.83070813155896868398567591001, −0.48860877890919594995270846709,
0.48860877890919594995270846709, 1.83070813155896868398567591001, 2.65862304203081840648378457415, 4.14744656494423976745085298384, 4.27925277160459172480525317097, 5.16793898148002160130131372872, 5.69254582856676851595706442169, 6.90293156584032551939407351109, 7.55404977772972906046212316126, 7.974925935330775927725964787199
