| L(s) = 1 | − 3.18·3-s − 14.3·5-s + 17.1·7-s − 16.8·9-s + 3.82·11-s − 4.03·13-s + 45.5·15-s + 17·17-s − 36.1·19-s − 54.5·21-s + 161.·23-s + 80.1·25-s + 139.·27-s + 94.5·29-s + 40.4·31-s − 12.1·33-s − 245.·35-s − 9.76·37-s + 12.8·39-s − 387.·41-s − 169.·43-s + 241.·45-s + 284.·47-s − 49.1·49-s − 54.0·51-s + 602.·53-s − 54.7·55-s + ⋯ |
| L(s) = 1 | − 0.612·3-s − 1.28·5-s + 0.925·7-s − 0.625·9-s + 0.104·11-s − 0.0861·13-s + 0.784·15-s + 0.242·17-s − 0.436·19-s − 0.566·21-s + 1.46·23-s + 0.641·25-s + 0.994·27-s + 0.605·29-s + 0.234·31-s − 0.0641·33-s − 1.18·35-s − 0.0434·37-s + 0.0527·39-s − 1.47·41-s − 0.602·43-s + 0.800·45-s + 0.883·47-s − 0.143·49-s − 0.148·51-s + 1.56·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 + 3.18T + 27T^{2} \) |
| 5 | \( 1 + 14.3T + 125T^{2} \) |
| 7 | \( 1 - 17.1T + 343T^{2} \) |
| 11 | \( 1 - 3.82T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.03T + 2.19e3T^{2} \) |
| 19 | \( 1 + 36.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 94.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 40.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 9.76T + 5.06e4T^{2} \) |
| 41 | \( 1 + 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 169.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 284.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 602.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 16.2T + 2.05e5T^{2} \) |
| 61 | \( 1 - 784.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 270.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 936.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.37e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 601.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 279.T + 9.12e5T^{2} \) |
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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
−8.664971590607185315587624476948, −8.415518473874262735421894520122, −7.39656493963079447908094306182, −6.67209584714652560309775271199, −5.44934788949114622194978711551, −4.79935876602284684994948985622, −3.85670918120957288409460360579, −2.74025231744973573091036195956, −1.13476693547516479492788611216, 0,
1.13476693547516479492788611216, 2.74025231744973573091036195956, 3.85670918120957288409460360579, 4.79935876602284684994948985622, 5.44934788949114622194978711551, 6.67209584714652560309775271199, 7.39656493963079447908094306182, 8.415518473874262735421894520122, 8.664971590607185315587624476948
