| L(s) = 1 | − 3-s + 2·5-s − 7-s + 9-s + 4·11-s − 13-s − 2·15-s + 6·17-s + 4·19-s + 21-s − 25-s − 27-s − 6·29-s + 8·31-s − 4·33-s − 2·35-s + 10·37-s + 39-s − 6·41-s − 4·43-s + 2·45-s − 4·47-s + 49-s − 6·51-s + 10·53-s + 8·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.338·35-s + 1.64·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 0.583·47-s + 1/7·49-s − 0.840·51-s + 1.37·53-s + 1.07·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112957301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112957301\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.354714758382660845311474798913, −7.51067746767874352362260855661, −6.79482302077840498445933050223, −6.03329386341551585804428431234, −5.63878669542096369971564363477, −4.75520608294001755631259806975, −3.79035699394372210432044760080, −2.96031750118234459450101408922, −1.74303235492918853958839595281, −0.900936951088735966242173950604,
0.900936951088735966242173950604, 1.74303235492918853958839595281, 2.96031750118234459450101408922, 3.79035699394372210432044760080, 4.75520608294001755631259806975, 5.63878669542096369971564363477, 6.03329386341551585804428431234, 6.79482302077840498445933050223, 7.51067746767874352362260855661, 8.354714758382660845311474798913
