| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 4·13-s − 15-s + 6·17-s + 2·21-s + 25-s + 27-s + 8·29-s + 31-s − 2·35-s + 4·37-s + 4·39-s + 10·41-s − 8·43-s − 45-s + 4·47-s − 3·49-s + 6·51-s − 14·53-s − 14·59-s − 6·61-s + 2·63-s − 4·65-s + 10·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.258·15-s + 1.45·17-s + 0.436·21-s + 1/5·25-s + 0.192·27-s + 1.48·29-s + 0.179·31-s − 0.338·35-s + 0.657·37-s + 0.640·39-s + 1.56·41-s − 1.21·43-s − 0.149·45-s + 0.583·47-s − 3/7·49-s + 0.840·51-s − 1.92·53-s − 1.82·59-s − 0.768·61-s + 0.251·63-s − 0.496·65-s + 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.125723720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.125723720\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.929051861249934832356481311560, −7.51756058099746396838226247290, −6.46193266592321770928633568410, −5.90043500550419334803503618108, −4.87649587064915316027668833442, −4.35753299899360647882796770062, −3.41546735722577944204540536846, −2.90951403003388796725691476457, −1.66543491238490246529369513531, −0.946664700471127729645202728930,
0.946664700471127729645202728930, 1.66543491238490246529369513531, 2.90951403003388796725691476457, 3.41546735722577944204540536846, 4.35753299899360647882796770062, 4.87649587064915316027668833442, 5.90043500550419334803503618108, 6.46193266592321770928633568410, 7.51756058099746396838226247290, 7.929051861249934832356481311560
