| L(s) = 1 | + 1.41·2-s − 1.41·3-s + 1.00·4-s − 5-s − 2.00·6-s + 1.00·9-s − 1.41·10-s − 1.41·12-s + 1.41·13-s + 1.41·15-s − 0.999·16-s + 1.41·18-s − 19-s − 1.00·20-s + 25-s + 2.00·26-s + 2.00·30-s − 1.41·32-s + 1.00·36-s − 1.41·37-s − 1.41·38-s − 2.00·39-s − 1.00·45-s + 1.41·48-s + 49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.41·3-s + 1.00·4-s − 5-s − 2.00·6-s + 1.00·9-s − 1.41·10-s − 1.41·12-s + 1.41·13-s + 1.41·15-s − 0.999·16-s + 1.41·18-s − 19-s − 1.00·20-s + 25-s + 2.00·26-s + 2.00·30-s − 1.41·32-s + 1.00·36-s − 1.41·37-s − 1.41·38-s − 2.00·39-s − 1.00·45-s + 1.41·48-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6531054055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6531054055\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.41T + T^{2} \) |
| 3 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.41T + T^{2} \) |
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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
−14.13944646882214871532234905578, −12.93875799995861054376414921190, −12.22315444399224238023174054159, −11.35478032774755308995244767083, −10.72883796175334370510374909405, −8.588273787705263002746602261300, −6.82424315532089692950431228435, −5.92355506824304704240004536396, −4.75709052287312937184999225316, −3.66881192896887252833915429072,
3.66881192896887252833915429072, 4.75709052287312937184999225316, 5.92355506824304704240004536396, 6.82424315532089692950431228435, 8.588273787705263002746602261300, 10.72883796175334370510374909405, 11.35478032774755308995244767083, 12.22315444399224238023174054159, 12.93875799995861054376414921190, 14.13944646882214871532234905578
