| L(s) = 1 | − 1.80·2-s + 1.24·3-s + 2.24·4-s − 2.24·6-s + 7-s − 2.24·8-s + 0.554·9-s + 2.80·12-s − 1.80·13-s − 1.80·14-s + 1.80·16-s − 0.445·17-s − 0.999·18-s − 0.445·19-s + 1.24·21-s + 1.24·23-s − 2.80·24-s + 25-s + 3.24·26-s − 0.554·27-s + 2.24·28-s − 1.00·32-s + 0.801·34-s + 1.24·36-s − 1.80·37-s + 0.801·38-s − 2.24·39-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 1.24·3-s + 2.24·4-s − 2.24·6-s + 7-s − 2.24·8-s + 0.554·9-s + 2.80·12-s − 1.80·13-s − 1.80·14-s + 1.80·16-s − 0.445·17-s − 0.999·18-s − 0.445·19-s + 1.24·21-s + 1.24·23-s − 2.80·24-s + 25-s + 3.24·26-s − 0.554·27-s + 2.24·28-s − 1.00·32-s + 0.801·34-s + 1.24·36-s − 1.80·37-s + 0.801·38-s − 2.24·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5307522865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5307522865\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.80T + T^{2} \) |
| 3 | \( 1 - 1.24T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 + 0.445T + T^{2} \) |
| 23 | \( 1 - 1.24T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + 0.445T + T^{2} \) |
| 47 | \( 1 + 1.80T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.80T + T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−11.65891880652348892549946795336, −10.78250559542412005593668909887, −9.864391262451580615867045765893, −8.986941469572290249075414720670, −8.460803414283570476536541005022, −7.57176915215757514066040403834, −6.91591547070030771755669009401, −4.90714365739911539367664818202, −2.86385095983606244584232727242, −1.89021135447327534675760110407,
1.89021135447327534675760110407, 2.86385095983606244584232727242, 4.90714365739911539367664818202, 6.91591547070030771755669009401, 7.57176915215757514066040403834, 8.460803414283570476536541005022, 8.986941469572290249075414720670, 9.864391262451580615867045765893, 10.78250559542412005593668909887, 11.65891880652348892549946795336
