| L(s) = 1 | − 4·7-s − 2·13-s − 17-s − 2·19-s − 4·23-s − 5·25-s + 6·29-s + 6·37-s + 2·41-s + 10·43-s + 2·47-s + 9·49-s − 10·53-s + 2·59-s − 12·61-s − 4·67-s − 8·71-s − 4·73-s + 2·89-s + 8·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 4·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.554·13-s − 0.242·17-s − 0.458·19-s − 0.834·23-s − 25-s + 1.11·29-s + 0.986·37-s + 0.312·41-s + 1.52·43-s + 0.291·47-s + 9/7·49-s − 1.37·53-s + 0.260·59-s − 1.53·61-s − 0.488·67-s − 0.949·71-s − 0.468·73-s + 0.211·89-s + 0.838·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.366·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.87984211397360, −12.47107209095703, −12.13900978868754, −11.65693106803505, −11.05364470491269, −10.54454702323938, −10.09502271606596, −9.777260203143412, −9.227639817890295, −8.997377326415830, −8.272464431951180, −7.645499626040465, −7.509832473788497, −6.634091734877015, −6.413727511753939, −5.923883267511615, −5.555823278992134, −4.650279486945461, −4.283986946526648, −3.847398100387731, −3.063307240731409, −2.766880745796894, −2.195378178972781, −1.453159922741699, −0.5662172813017881, 0,
0.5662172813017881, 1.453159922741699, 2.195378178972781, 2.766880745796894, 3.063307240731409, 3.847398100387731, 4.283986946526648, 4.650279486945461, 5.555823278992134, 5.923883267511615, 6.413727511753939, 6.634091734877015, 7.509832473788497, 7.645499626040465, 8.272464431951180, 8.997377326415830, 9.227639817890295, 9.777260203143412, 10.09502271606596, 10.54454702323938, 11.05364470491269, 11.65693106803505, 12.13900978868754, 12.47107209095703, 12.87984211397360
