| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 4·11-s + 2·13-s + 16-s + 2·17-s − 4·19-s + 20-s + 4·22-s + 25-s + 2·26-s + 6·29-s − 4·31-s + 32-s + 2·34-s − 37-s − 4·38-s + 40-s + 6·41-s + 4·43-s + 4·44-s + 8·47-s − 7·49-s + 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 0.164·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + 0.603·44-s + 1.16·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.701468404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.701468404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.682599430842842951185945495515, −7.81059324472633605684417031104, −6.87493338522800143592874040603, −6.27790394394954895549345060581, −5.72116256637183355409115684013, −4.69247680915447547060653501609, −4.00369031444618358374700555311, −3.17732882969520207000389302979, −2.10151323526117516194563984785, −1.12328850500342707125804122399,
1.12328850500342707125804122399, 2.10151323526117516194563984785, 3.17732882969520207000389302979, 4.00369031444618358374700555311, 4.69247680915447547060653501609, 5.72116256637183355409115684013, 6.27790394394954895549345060581, 6.87493338522800143592874040603, 7.81059324472633605684417031104, 8.682599430842842951185945495515
