| L(s) = 1 | + 20.1·5-s + 19.6·7-s + 13.5·11-s + 13·13-s − 116.·17-s − 68.1·19-s + 122.·23-s + 280.·25-s − 204.·29-s + 194.·31-s + 396.·35-s − 142.·37-s + 175.·41-s + 219.·43-s + 236.·47-s + 45.0·49-s + 628.·53-s + 273.·55-s + 446.·59-s + 224.·61-s + 261.·65-s − 165.·67-s + 902.·71-s − 15.1·73-s + 266.·77-s − 670.·79-s + 1.04e3·83-s + ⋯ |
| L(s) = 1 | + 1.80·5-s + 1.06·7-s + 0.371·11-s + 0.277·13-s − 1.66·17-s − 0.822·19-s + 1.10·23-s + 2.24·25-s − 1.30·29-s + 1.12·31-s + 1.91·35-s − 0.634·37-s + 0.666·41-s + 0.778·43-s + 0.733·47-s + 0.131·49-s + 1.62·53-s + 0.669·55-s + 0.984·59-s + 0.471·61-s + 0.499·65-s − 0.301·67-s + 1.50·71-s − 0.0242·73-s + 0.395·77-s − 0.954·79-s + 1.37·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.971221979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.971221979\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 20.1T + 125T^{2} \) |
| 7 | \( 1 - 19.6T + 343T^{2} \) |
| 11 | \( 1 - 13.5T + 1.33e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 68.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 175.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 219.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 236.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 628.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 224.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 165.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 902.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 15.1T + 3.89e5T^{2} \) |
| 79 | \( 1 + 670.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 562.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.64e3T + 9.12e5T^{2} \) |
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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
−8.979612251648556713387301904354, −8.342495738358660732962068077466, −7.07314262565942452783780317368, −6.46761007700977991576663712401, −5.64407989849702682027840272393, −4.93482364942701458627533193608, −4.07033430036915002789681059487, −2.48043625507977346678838467219, −1.98055245855744336167724019081, −0.979320264978208541398548136926,
0.979320264978208541398548136926, 1.98055245855744336167724019081, 2.48043625507977346678838467219, 4.07033430036915002789681059487, 4.93482364942701458627533193608, 5.64407989849702682027840272393, 6.46761007700977991576663712401, 7.07314262565942452783780317368, 8.342495738358660732962068077466, 8.979612251648556713387301904354
