| L(s) = 1 | − 1.17·3-s − 5·5-s + 7·7-s − 25.6·9-s − 52.0·11-s − 63.8·13-s + 5.89·15-s + 51.2·17-s − 56.1·19-s − 8.25·21-s + 97.1·23-s + 25·25-s + 62.0·27-s − 300.·29-s − 181.·31-s + 61.3·33-s − 35·35-s + 95.8·37-s + 75.2·39-s + 247.·41-s + 414.·43-s + 128.·45-s + 24.1·47-s + 49·49-s − 60.4·51-s + 399.·53-s + 260.·55-s + ⋯ |
| L(s) = 1 | − 0.226·3-s − 0.447·5-s + 0.377·7-s − 0.948·9-s − 1.42·11-s − 1.36·13-s + 0.101·15-s + 0.730·17-s − 0.677·19-s − 0.0857·21-s + 0.880·23-s + 0.200·25-s + 0.442·27-s − 1.92·29-s − 1.04·31-s + 0.323·33-s − 0.169·35-s + 0.425·37-s + 0.308·39-s + 0.941·41-s + 1.47·43-s + 0.424·45-s + 0.0748·47-s + 0.142·49-s − 0.165·51-s + 1.03·53-s + 0.638·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7793979968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7793979968\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 + 1.17T + 27T^{2} \) |
| 11 | \( 1 + 52.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 63.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 56.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 97.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 300.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 95.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 247.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 414.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 24.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 399.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 353.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 586.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 413.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 36.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 472.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 267.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 336.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 375.T + 9.12e5T^{2} \) |
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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
−9.409243020241471684020993412446, −8.596362144164793878384694694715, −7.57661669815135322186250658208, −7.35743606272283007402295819854, −5.74892286692264948989389715902, −5.33057422099453134926627830931, −4.32856104167790956759299976728, −3.03735206779802659597421475108, −2.21809155055854438924007355552, −0.43657583895959600688413584406,
0.43657583895959600688413584406, 2.21809155055854438924007355552, 3.03735206779802659597421475108, 4.32856104167790956759299976728, 5.33057422099453134926627830931, 5.74892286692264948989389715902, 7.35743606272283007402295819854, 7.57661669815135322186250658208, 8.596362144164793878384694694715, 9.409243020241471684020993412446
