| L(s) = 1 | + 10.7·2-s − 12.0·4-s + 420.·7-s − 1.50e3·8-s + 6.82e3·11-s − 1.01e4·13-s + 4.52e3·14-s − 1.47e4·16-s + 1.56e4·17-s − 6.86e3·19-s + 7.35e4·22-s − 2.92e4·23-s − 1.09e5·26-s − 5.04e3·28-s + 2.55e4·29-s + 8.21e4·31-s + 3.46e4·32-s + 1.68e5·34-s + 2.23e5·37-s − 7.38e4·38-s + 5.33e5·41-s + 7.08e5·43-s − 8.19e4·44-s − 3.14e5·46-s + 5.82e3·47-s − 6.47e5·49-s + 1.21e5·52-s + ⋯ |
| L(s) = 1 | + 0.951·2-s − 0.0937·4-s + 0.462·7-s − 1.04·8-s + 1.54·11-s − 1.28·13-s + 0.440·14-s − 0.897·16-s + 0.774·17-s − 0.229·19-s + 1.47·22-s − 0.500·23-s − 1.21·26-s − 0.0433·28-s + 0.194·29-s + 0.495·31-s + 0.186·32-s + 0.736·34-s + 0.725·37-s − 0.218·38-s + 1.20·41-s + 1.35·43-s − 0.145·44-s − 0.476·46-s + 0.00818·47-s − 0.785·49-s + 0.120·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.237151933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.237151933\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 10.7T + 128T^{2} \) |
| 7 | \( 1 - 420.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.82e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.01e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.56e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.86e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.92e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.55e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 8.21e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.33e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.82e3T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.89e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.43e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.38e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.81e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.48e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.05e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.60e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.20e7T + 8.07e13T^{2} \) |
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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
−11.37898612035935731739709917058, −9.881275197971010634773984026963, −9.148966003804422565215370496406, −7.948342240086720147068115288917, −6.69490019097783677365783032328, −5.66023618262263528385180933101, −4.59285907259561457301410990251, −3.79931019734209690930737676314, −2.42659148864675242832620016584, −0.829842706369900636527501485858,
0.829842706369900636527501485858, 2.42659148864675242832620016584, 3.79931019734209690930737676314, 4.59285907259561457301410990251, 5.66023618262263528385180933101, 6.69490019097783677365783032328, 7.948342240086720147068115288917, 9.148966003804422565215370496406, 9.881275197971010634773984026963, 11.37898612035935731739709917058
