| L(s) = 1 | + 3-s + 3·5-s + 7-s + 9-s − 11-s + 13-s + 3·15-s − 17-s − 6·19-s + 21-s + 2·23-s + 4·25-s + 27-s − 2·29-s − 33-s + 3·35-s + 5·37-s + 39-s + 4·41-s + 9·43-s + 3·45-s + 49-s − 51-s + 11·53-s − 3·55-s − 6·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.774·15-s − 0.242·17-s − 1.37·19-s + 0.218·21-s + 0.417·23-s + 4/5·25-s + 0.192·27-s − 0.371·29-s − 0.174·33-s + 0.507·35-s + 0.821·37-s + 0.160·39-s + 0.624·41-s + 1.37·43-s + 0.447·45-s + 1/7·49-s − 0.140·51-s + 1.51·53-s − 0.404·55-s − 0.794·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.476984177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.476984177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−8.230181233163377072391206556685, −7.46789639665146044636445370509, −6.61533913687974552980184755280, −6.01170978697682863588758156995, −5.30740827986631055992699318293, −4.46932263830009852693651038810, −3.66833515921551500358074348792, −2.37599584843001900977226187836, −2.21361673138795484971928240020, −0.994417332982276826321481545703,
0.994417332982276826321481545703, 2.21361673138795484971928240020, 2.37599584843001900977226187836, 3.66833515921551500358074348792, 4.46932263830009852693651038810, 5.30740827986631055992699318293, 6.01170978697682863588758156995, 6.61533913687974552980184755280, 7.46789639665146044636445370509, 8.230181233163377072391206556685
