| L(s) = 1 | + 3·5-s + 3·11-s + 4·13-s − 4·19-s + 4·25-s − 9·29-s − 31-s + 8·37-s + 10·43-s + 6·47-s + 3·53-s + 9·55-s − 3·59-s + 10·61-s + 12·65-s + 10·67-s − 6·71-s − 2·73-s + 79-s + 9·83-s + 6·89-s − 12·95-s + 97-s − 18·101-s + 8·103-s − 3·107-s + 14·109-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.904·11-s + 1.10·13-s − 0.917·19-s + 4/5·25-s − 1.67·29-s − 0.179·31-s + 1.31·37-s + 1.52·43-s + 0.875·47-s + 0.412·53-s + 1.21·55-s − 0.390·59-s + 1.28·61-s + 1.48·65-s + 1.22·67-s − 0.712·71-s − 0.234·73-s + 0.112·79-s + 0.987·83-s + 0.635·89-s − 1.23·95-s + 0.101·97-s − 1.79·101-s + 0.788·103-s − 0.290·107-s + 1.34·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.091382749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.091382749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.973598604422608856711125899156, −7.12684258637692642405801706265, −6.30981704578337725497607593309, −5.97708585339505464195745997258, −5.33778120225130226806025517412, −4.21075854793680523594236828144, −3.71168889085604917791465076443, −2.50214901196819952049505879216, −1.83818484293011714855169652362, −0.942551933080821992746372320770,
0.942551933080821992746372320770, 1.83818484293011714855169652362, 2.50214901196819952049505879216, 3.71168889085604917791465076443, 4.21075854793680523594236828144, 5.33778120225130226806025517412, 5.97708585339505464195745997258, 6.30981704578337725497607593309, 7.12684258637692642405801706265, 7.973598604422608856711125899156
