L(s) = 1 | − 3-s + 3.70·5-s − 1.70·7-s + 9-s − 1.70·11-s − 6·13-s − 3.70·15-s + 3.70·17-s + 19-s + 1.70·21-s − 4·23-s + 8.70·25-s − 27-s − 2·29-s + 3.40·31-s + 1.70·33-s − 6.29·35-s − 9.40·37-s + 6·39-s + 9.40·41-s − 9.10·43-s + 3.70·45-s − 5.70·47-s − 4.10·49-s − 3.70·51-s + 6·53-s − 6.29·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.65·5-s − 0.643·7-s + 0.333·9-s − 0.513·11-s − 1.66·13-s − 0.955·15-s + 0.897·17-s + 0.229·19-s + 0.371·21-s − 0.834·23-s + 1.74·25-s − 0.192·27-s − 0.371·29-s + 0.611·31-s + 0.296·33-s − 1.06·35-s − 1.54·37-s + 0.960·39-s + 1.46·41-s − 1.38·43-s + 0.551·45-s − 0.831·47-s − 0.586·49-s − 0.518·51-s + 0.824·53-s − 0.849·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 + 1.70T + 7T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 - 9.40T + 41T^{2} \) |
| 43 | \( 1 + 9.10T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 7.70T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 0.298T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.077342206820065567765041244831, −7.27658138751254972395870708021, −6.55175999568645575358227596781, −5.84839698076711898294095629094, −5.30652218538457381897603655384, −4.65401599809122212921915955933, −3.24659206653404389613743240261, −2.42865711470263501591328849768, −1.53915144002156966880209219833, 0,
1.53915144002156966880209219833, 2.42865711470263501591328849768, 3.24659206653404389613743240261, 4.65401599809122212921915955933, 5.30652218538457381897603655384, 5.84839698076711898294095629094, 6.55175999568645575358227596781, 7.27658138751254972395870708021, 8.077342206820065567765041244831