| L(s) = 1 | − 20.2·2-s − 103.·4-s + 4.01e3·7-s + 1.24e4·8-s + 4.21e4·11-s − 1.23e5·13-s − 8.10e4·14-s − 1.98e5·16-s + 3.19e5·17-s + 1.08e6·19-s − 8.50e5·22-s + 1.50e6·23-s + 2.49e6·26-s − 4.16e5·28-s + 2.62e6·29-s + 3.27e6·31-s − 2.36e6·32-s − 6.46e6·34-s + 2.51e6·37-s − 2.19e7·38-s − 2.95e7·41-s + 1.42e7·43-s − 4.37e6·44-s − 3.04e7·46-s + 1.35e6·47-s − 2.42e7·49-s + 1.28e7·52-s + ⋯ |
| L(s) = 1 | − 0.892·2-s − 0.202·4-s + 0.631·7-s + 1.07·8-s + 0.867·11-s − 1.20·13-s − 0.563·14-s − 0.755·16-s + 0.929·17-s + 1.91·19-s − 0.774·22-s + 1.12·23-s + 1.07·26-s − 0.128·28-s + 0.688·29-s + 0.635·31-s − 0.399·32-s − 0.829·34-s + 0.220·37-s − 1.70·38-s − 1.63·41-s + 0.635·43-s − 0.175·44-s − 1.00·46-s + 0.0404·47-s − 0.601·49-s + 0.243·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.536538579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536538579\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 20.2T + 512T^{2} \) |
| 7 | \( 1 - 4.01e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.21e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.23e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.19e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.08e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.50e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.62e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.27e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.51e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.95e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.35e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.73e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.48e6T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.11e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.94e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.82e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.55e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.48e6T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.25e8T + 7.60e17T^{2} \) |
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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
−10.27563387367475612649701542560, −9.621220937354214826921331684204, −8.768859186501591311851368775154, −7.72725622816351010149287216638, −7.06115587040492214588649187898, −5.35752415606959714442212605820, −4.54802462292572869668550850060, −3.08622933756801272134121488349, −1.48979443769921550636481583295, −0.74958622579217496663058909125,
0.74958622579217496663058909125, 1.48979443769921550636481583295, 3.08622933756801272134121488349, 4.54802462292572869668550850060, 5.35752415606959714442212605820, 7.06115587040492214588649187898, 7.72725622816351010149287216638, 8.768859186501591311851368775154, 9.621220937354214826921331684204, 10.27563387367475612649701542560
