| L(s) = 1 | − 5·2-s − 4·3-s + 17·4-s + 20·6-s + 32·7-s − 45·8-s − 11·9-s − 12·11-s − 68·12-s + 42·13-s − 160·14-s + 89·16-s − 114·17-s + 55·18-s + 19·19-s − 128·21-s + 60·22-s − 160·23-s + 180·24-s − 210·26-s + 152·27-s + 544·28-s + 214·29-s − 144·31-s − 85·32-s + 48·33-s + 570·34-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 0.769·3-s + 17/8·4-s + 1.36·6-s + 1.72·7-s − 1.98·8-s − 0.407·9-s − 0.328·11-s − 1.63·12-s + 0.896·13-s − 3.05·14-s + 1.39·16-s − 1.62·17-s + 0.720·18-s + 0.229·19-s − 1.33·21-s + 0.581·22-s − 1.45·23-s + 1.53·24-s − 1.58·26-s + 1.08·27-s + 3.67·28-s + 1.37·29-s − 0.834·31-s − 0.469·32-s + 0.253·33-s + 2.87·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - p T \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 23 | \( 1 + 160 T + p^{3} T^{2} \) |
| 29 | \( 1 - 214 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 94 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 308 T + p^{3} T^{2} \) |
| 47 | \( 1 + 184 T + p^{3} T^{2} \) |
| 53 | \( 1 - 274 T + p^{3} T^{2} \) |
| 59 | \( 1 - 276 T + p^{3} T^{2} \) |
| 61 | \( 1 + 826 T + p^{3} T^{2} \) |
| 67 | \( 1 + 52 T + p^{3} T^{2} \) |
| 71 | \( 1 + 344 T + p^{3} T^{2} \) |
| 73 | \( 1 - 166 T + p^{3} T^{2} \) |
| 79 | \( 1 + 688 T + p^{3} T^{2} \) |
| 83 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 89 | \( 1 - 1578 T + p^{3} T^{2} \) |
| 97 | \( 1 + 786 T + p^{3} T^{2} \) |
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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
−10.41081659946184676121033430787, −8.987730233220585274147128282699, −8.443516107976756812804201045815, −7.75970730170505739224459858302, −6.64708937038768895819795896514, −5.70376526000753830502370493900, −4.47887911858080895691550436600, −2.34792238038510487883997445561, −1.30370790381281940557442809246, 0,
1.30370790381281940557442809246, 2.34792238038510487883997445561, 4.47887911858080895691550436600, 5.70376526000753830502370493900, 6.64708937038768895819795896514, 7.75970730170505739224459858302, 8.443516107976756812804201045815, 8.987730233220585274147128282699, 10.41081659946184676121033430787
