| L(s) = 1 | − 3.85·3-s − 0.490i·5-s − 2.64i·7-s + 5.87·9-s + 15.5·11-s + 3.50i·13-s + 1.89i·15-s − 24.1·17-s − 3.56·19-s + 10.2i·21-s + 19.5i·23-s + 24.7·25-s + 12.0·27-s − 10.9i·29-s + 21.1i·31-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 0.0980i·5-s − 0.377i·7-s + 0.652·9-s + 1.41·11-s + 0.269i·13-s + 0.126i·15-s − 1.42·17-s − 0.187·19-s + 0.485i·21-s + 0.851i·23-s + 0.990·25-s + 0.446·27-s − 0.378i·29-s + 0.683i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9472799216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9472799216\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 3 | \( 1 + 3.85T + 9T^{2} \) |
| 5 | \( 1 + 0.490iT - 25T^{2} \) |
| 11 | \( 1 - 15.5T + 121T^{2} \) |
| 13 | \( 1 - 3.50iT - 169T^{2} \) |
| 17 | \( 1 + 24.1T + 289T^{2} \) |
| 19 | \( 1 + 3.56T + 361T^{2} \) |
| 23 | \( 1 - 19.5iT - 529T^{2} \) |
| 29 | \( 1 + 10.9iT - 841T^{2} \) |
| 31 | \( 1 - 21.1iT - 961T^{2} \) |
| 37 | \( 1 + 58.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 54.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 35.6T + 1.84e3T^{2} \) |
| 47 | \( 1 - 64.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 87.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 66.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 16.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 21.2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 64.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 99.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.03T + 6.88e3T^{2} \) |
| 89 | \( 1 - 23.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 171.T + 9.40e3T^{2} \) |
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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
−9.119165012446371805715480994564, −8.634021727329050129869491172175, −7.26899444505754266950215189072, −6.66199072757533575926119926906, −6.13896108669361908585331856992, −5.08880448737239665975035679760, −4.41894862336378887588543230194, −3.49756349165512377120508565206, −1.88237551733600271315460265742, −0.78200009438667895537188212725,
0.43044475892864852544744528629, 1.67518599991345412715659400034, 3.01437725144366112151198099778, 4.29005785823942348147068115718, 4.91271582194253581628532196542, 5.88608544985241345449568862336, 6.64421191965548181919590509699, 6.90187669561246043707172013576, 8.488265963477378338622622393610, 8.844115847349062654511253467446
