| L(s) = 1 | + 8.54·3-s − 13.1·5-s − 19.9·7-s + 45.9·9-s − 25.7·11-s + 81.5·13-s − 112.·15-s + 115.·17-s − 19·19-s − 170.·21-s − 167.·23-s + 48.1·25-s + 161.·27-s − 251.·29-s + 189.·31-s − 220.·33-s + 262.·35-s − 88.3·37-s + 696.·39-s − 125.·41-s − 459.·43-s − 604.·45-s − 509.·47-s + 55.6·49-s + 986.·51-s + 604.·53-s + 339.·55-s + ⋯ |
| L(s) = 1 | + 1.64·3-s − 1.17·5-s − 1.07·7-s + 1.70·9-s − 0.707·11-s + 1.73·13-s − 1.93·15-s + 1.64·17-s − 0.229·19-s − 1.77·21-s − 1.51·23-s + 0.385·25-s + 1.15·27-s − 1.61·29-s + 1.09·31-s − 1.16·33-s + 1.26·35-s − 0.392·37-s + 2.85·39-s − 0.479·41-s − 1.62·43-s − 2.00·45-s − 1.58·47-s + 0.162·49-s + 2.70·51-s + 1.56·53-s + 0.832·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 8.54T + 27T^{2} \) |
| 5 | \( 1 + 13.1T + 125T^{2} \) |
| 7 | \( 1 + 19.9T + 343T^{2} \) |
| 11 | \( 1 + 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 81.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 115.T + 4.91e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 251.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 88.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 459.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 509.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 604.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 271.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 397.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 762.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 108.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 230.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 412.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−8.651246757448325320751602265334, −8.175894187858316777294401937200, −7.69067621198875833580646061592, −6.68140687988901054002520825349, −5.61763693097654337363625155292, −3.98844575218101081940878940184, −3.58628649057336736843301034191, −2.97234811028860268322591843453, −1.57385132986058283642785428430, 0,
1.57385132986058283642785428430, 2.97234811028860268322591843453, 3.58628649057336736843301034191, 3.98844575218101081940878940184, 5.61763693097654337363625155292, 6.68140687988901054002520825349, 7.69067621198875833580646061592, 8.175894187858316777294401937200, 8.651246757448325320751602265334
