| L(s) = 1 | + 5-s + 2·7-s + 6·13-s + 7·17-s + 7·19-s − 7·23-s + 25-s − 6·29-s − 3·31-s + 2·35-s + 6·37-s + 4·41-s + 8·43-s + 4·47-s − 3·49-s + 5·53-s + 6·59-s + 3·61-s + 6·65-s − 10·67-s − 12·71-s + 16·73-s − 79-s + 9·83-s + 7·85-s − 4·89-s + 12·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 1.66·13-s + 1.69·17-s + 1.60·19-s − 1.45·23-s + 1/5·25-s − 1.11·29-s − 0.538·31-s + 0.338·35-s + 0.986·37-s + 0.624·41-s + 1.21·43-s + 0.583·47-s − 3/7·49-s + 0.686·53-s + 0.781·59-s + 0.384·61-s + 0.744·65-s − 1.22·67-s − 1.42·71-s + 1.87·73-s − 0.112·79-s + 0.987·83-s + 0.759·85-s − 0.423·89-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.246566809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.246566809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.81239295473233379077784529073, −7.28348193923657612111973074087, −6.13804575160525039712766109347, −5.69568468995998131868803278612, −5.26148488082895507630327656550, −4.05297786308371571243287583478, −3.62349445900645266899467343816, −2.62933892402557776457814827284, −1.54270187960776649729784848340, −1.00305233293312481201857581927,
1.00305233293312481201857581927, 1.54270187960776649729784848340, 2.62933892402557776457814827284, 3.62349445900645266899467343816, 4.05297786308371571243287583478, 5.26148488082895507630327656550, 5.69568468995998131868803278612, 6.13804575160525039712766109347, 7.28348193923657612111973074087, 7.81239295473233379077784529073
