| L(s) = 1 | − 3.20·3-s + 7.26·9-s + 4.20·11-s + 0.204·13-s − 5.06·17-s + 1.06·19-s + 2.14·23-s − 13.6·27-s − 7.47·29-s − 8.47·31-s − 13.4·33-s − 10.6·37-s − 0.654·39-s + 10.5·41-s + 8.26·43-s + 3.26·47-s + 16.2·51-s + 5.67·53-s − 3.40·57-s + 1.20·59-s + 1.65·61-s − 12.4·67-s − 6.85·69-s − 0.591·71-s − 4·73-s + 6.54·79-s + 22.0·81-s + ⋯ |
| L(s) = 1 | − 1.85·3-s + 2.42·9-s + 1.26·11-s + 0.0566·13-s − 1.22·17-s + 0.244·19-s + 0.446·23-s − 2.63·27-s − 1.38·29-s − 1.52·31-s − 2.34·33-s − 1.74·37-s − 0.104·39-s + 1.64·41-s + 1.26·43-s + 0.476·47-s + 2.27·51-s + 0.779·53-s − 0.451·57-s + 0.156·59-s + 0.211·61-s − 1.51·67-s − 0.825·69-s − 0.0701·71-s − 0.468·73-s + 0.736·79-s + 2.44·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.20T + 3T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 - 0.204T + 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 - 1.06T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 8.47T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 8.26T + 43T^{2} \) |
| 47 | \( 1 - 3.26T + 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 - 1.20T + 59T^{2} \) |
| 61 | \( 1 - 1.65T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 0.591T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 6.54T + 79T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 + 9.26T + 89T^{2} \) |
| 97 | \( 1 - 1.33T + 97T^{2} \) |
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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
−7.38326589098707973131463622273, −7.15933245211294192484394288746, −6.30564278188283802954254347446, −5.79320526326866992594591918765, −5.10519408038735440619761125044, −4.24805233955196192141229409764, −3.72611543418915920743764371028, −2.05180450179778004754934583423, −1.13338499387032574266983770810, 0,
1.13338499387032574266983770810, 2.05180450179778004754934583423, 3.72611543418915920743764371028, 4.24805233955196192141229409764, 5.10519408038735440619761125044, 5.79320526326866992594591918765, 6.30564278188283802954254347446, 7.15933245211294192484394288746, 7.38326589098707973131463622273
