L(s) = 1 | + 3-s − 4·7-s + 9-s − 13-s − 5·19-s − 4·21-s + 27-s + 3·29-s + 4·31-s + 7·37-s − 39-s + 3·41-s + 2·43-s + 9·47-s + 9·49-s − 9·53-s − 5·57-s − 6·59-s + 8·61-s − 4·63-s + 5·67-s + 3·71-s + 4·73-s − 11·79-s + 81-s − 6·83-s + 3·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 1.14·19-s − 0.872·21-s + 0.192·27-s + 0.557·29-s + 0.718·31-s + 1.15·37-s − 0.160·39-s + 0.468·41-s + 0.304·43-s + 1.31·47-s + 9/7·49-s − 1.23·53-s − 0.662·57-s − 0.781·59-s + 1.02·61-s − 0.503·63-s + 0.610·67-s + 0.356·71-s + 0.468·73-s − 1.23·79-s + 1/9·81-s − 0.658·83-s + 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.09793182699033, −15.70275471840298, −15.32521153016333, −14.48273778530153, −14.17812630052492, −13.35391188277604, −12.99422462112745, −12.51441110483801, −12.00597696594013, −11.10357669564575, −10.53608143059973, −9.874821125277884, −9.542338247348030, −8.885473896736737, −8.315312491543125, −7.610067293275028, −6.950718013189171, −6.323299685874655, −5.956694746323994, −4.881094293332312, −4.173561952074681, −3.588136451985782, −2.718314028682113, −2.385137556069775, −1.076740622380979, 0,
1.076740622380979, 2.385137556069775, 2.718314028682113, 3.588136451985782, 4.173561952074681, 4.881094293332312, 5.956694746323994, 6.323299685874655, 6.950718013189171, 7.610067293275028, 8.315312491543125, 8.885473896736737, 9.542338247348030, 9.874821125277884, 10.53608143059973, 11.10357669564575, 12.00597696594013, 12.51441110483801, 12.99422462112745, 13.35391188277604, 14.17812630052492, 14.48273778530153, 15.32521153016333, 15.70275471840298, 16.09793182699033