| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 3·11-s + 13-s + 14-s + 16-s − 19-s − 3·22-s − 6·23-s − 26-s − 28-s + 6·29-s − 4·31-s − 32-s − 2·37-s + 38-s + 3·41-s + 43-s + 3·44-s + 6·46-s − 9·47-s + 49-s + 52-s + 3·53-s + 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.904·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.229·19-s − 0.639·22-s − 1.25·23-s − 0.196·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.328·37-s + 0.162·38-s + 0.468·41-s + 0.152·43-s + 0.452·44-s + 0.884·46-s − 1.31·47-s + 1/7·49-s + 0.138·52-s + 0.412·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−7.39581636329443425090822460603, −6.66534025920333001674860917569, −6.22076158767313197351294759751, −5.50134345395059572514577625796, −4.43930319443907628266596680288, −3.77434859405461778758856901194, −2.96231013236206843868546452826, −2.00683669399701423254629215986, −1.18058438877120625235240269085, 0,
1.18058438877120625235240269085, 2.00683669399701423254629215986, 2.96231013236206843868546452826, 3.77434859405461778758856901194, 4.43930319443907628266596680288, 5.50134345395059572514577625796, 6.22076158767313197351294759751, 6.66534025920333001674860917569, 7.39581636329443425090822460603
