| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 7-s − 8-s + 9-s + 2·10-s − 4·11-s + 12-s + 13-s + 14-s − 2·15-s + 16-s − 2·17-s − 18-s − 4·19-s − 2·20-s − 21-s + 4·22-s − 4·23-s − 24-s − 25-s − 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.218·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−10.33940918099112086508153275865, −9.407728550276739147993380802554, −8.419058620635545585592668255564, −7.919520600930853256251460201426, −7.05482338122635591122480401032, −5.93037987998780819768001729788, −4.44015518719984939594103992774, −3.34291279159901837009808602924, −2.15328177975657163080119509125, 0,
2.15328177975657163080119509125, 3.34291279159901837009808602924, 4.44015518719984939594103992774, 5.93037987998780819768001729788, 7.05482338122635591122480401032, 7.919520600930853256251460201426, 8.419058620635545585592668255564, 9.407728550276739147993380802554, 10.33940918099112086508153275865
