| L(s) = 1 | − 21.3·2-s − 45.8·3-s + 329.·4-s + 980.·6-s + 104.·7-s − 4.31e3·8-s − 85.1·9-s − 6.23e3·11-s − 1.51e4·12-s − 7.88e3·13-s − 2.23e3·14-s + 5.01e4·16-s − 3.44e4·17-s + 1.82e3·18-s + 1.97e4·19-s − 4.79e3·21-s + 1.33e5·22-s − 8.97e4·23-s + 1.97e5·24-s + 1.68e5·26-s + 1.04e5·27-s + 3.44e4·28-s − 1.08e5·29-s + 2.66e5·31-s − 5.21e5·32-s + 2.86e5·33-s + 7.37e5·34-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 0.980·3-s + 2.57·4-s + 1.85·6-s + 0.115·7-s − 2.98·8-s − 0.0389·9-s − 1.41·11-s − 2.52·12-s − 0.995·13-s − 0.217·14-s + 3.06·16-s − 1.70·17-s + 0.0736·18-s + 0.660·19-s − 0.112·21-s + 2.67·22-s − 1.53·23-s + 2.92·24-s + 1.88·26-s + 1.01·27-s + 0.296·28-s − 0.823·29-s + 1.60·31-s − 2.81·32-s + 1.38·33-s + 3.21·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 21.3T + 128T^{2} \) |
| 3 | \( 1 + 45.8T + 2.18e3T^{2} \) |
| 7 | \( 1 - 104.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.23e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.88e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.44e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.97e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.97e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.08e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.66e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.36e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.95e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.04e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.85e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.09e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.47e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.79e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.13e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.15e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.11e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.01e4T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.29e6T + 8.07e13T^{2} \) |
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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
−9.166774599590654050460307543462, −8.106440769277981050236061504192, −7.59663762712120066731179160088, −6.56867284158966206828000489463, −5.82562048497712608382242144141, −4.72780163817623741599915155648, −2.72697215264274292034408496354, −2.06474776121088026367608306921, −0.61473915639404193927196381865, 0,
0.61473915639404193927196381865, 2.06474776121088026367608306921, 2.72697215264274292034408496354, 4.72780163817623741599915155648, 5.82562048497712608382242144141, 6.56867284158966206828000489463, 7.59663762712120066731179160088, 8.106440769277981050236061504192, 9.166774599590654050460307543462
