| L(s) = 1 | + 150.·2-s + 5.90e4·3-s − 2.07e6·4-s + 8.87e6·6-s − 6.41e7·7-s − 6.26e8·8-s + 3.48e9·9-s + 8.03e10·11-s − 1.22e11·12-s − 2.97e11·13-s − 9.63e9·14-s + 4.25e12·16-s + 2.29e12·17-s + 5.23e11·18-s − 1.07e13·19-s − 3.78e12·21-s + 1.20e13·22-s − 2.35e13·23-s − 3.70e13·24-s − 4.46e13·26-s + 2.05e14·27-s + 1.32e14·28-s + 1.90e14·29-s + 1.06e15·31-s + 1.95e15·32-s + 4.74e15·33-s + 3.44e14·34-s + ⋯ |
| L(s) = 1 | + 0.103·2-s + 0.577·3-s − 0.989·4-s + 0.0599·6-s − 0.0857·7-s − 0.206·8-s + 0.333·9-s + 0.933·11-s − 0.571·12-s − 0.598·13-s − 0.00889·14-s + 0.967·16-s + 0.275·17-s + 0.0345·18-s − 0.403·19-s − 0.0495·21-s + 0.0968·22-s − 0.118·23-s − 0.119·24-s − 0.0620·26-s + 0.192·27-s + 0.0848·28-s + 0.0841·29-s + 0.233·31-s + 0.306·32-s + 0.538·33-s + 0.0285·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.068860851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.068860851\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 150.T + 2.09e6T^{2} \) |
| 7 | \( 1 + 6.41e7T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.03e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 2.97e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.29e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.07e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.35e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.90e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 1.06e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.21e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 6.62e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.50e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.16e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.83e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.51e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.80e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 9.29e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.74e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 6.80e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 5.70e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.10e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 4.12e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 9.12e20T + 5.27e41T^{2} \) |
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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
−10.34473271992461269391125821589, −9.418872385132807437553836654408, −8.674497641157001292263495786009, −7.60570132092546921115919183512, −6.31458724345015541206100730436, −4.98444483671620355584827277509, −4.05437382120232729888620673376, −3.12814418687808004162451347569, −1.74585212767962334278938758073, −0.58406991789623872415746253991,
0.58406991789623872415746253991, 1.74585212767962334278938758073, 3.12814418687808004162451347569, 4.05437382120232729888620673376, 4.98444483671620355584827277509, 6.31458724345015541206100730436, 7.60570132092546921115919183512, 8.674497641157001292263495786009, 9.418872385132807437553836654408, 10.34473271992461269391125821589
