| L(s) = 1 | + 8.18·3-s + 4.03·5-s − 15.4·7-s + 39.9·9-s − 4.43·11-s − 27.5·13-s + 33.0·15-s + 44.6·19-s − 126.·21-s + 115.·23-s − 108.·25-s + 105.·27-s − 304.·29-s − 105.·31-s − 36.2·33-s − 62.1·35-s − 269.·37-s − 225.·39-s − 206.·41-s − 152.·43-s + 161.·45-s + 486.·47-s − 105.·49-s + 298.·53-s − 17.8·55-s + 365.·57-s − 535.·59-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 0.360·5-s − 0.832·7-s + 1.47·9-s − 0.121·11-s − 0.588·13-s + 0.568·15-s + 0.538·19-s − 1.31·21-s + 1.04·23-s − 0.869·25-s + 0.753·27-s − 1.95·29-s − 0.610·31-s − 0.191·33-s − 0.300·35-s − 1.19·37-s − 0.926·39-s − 0.787·41-s − 0.541·43-s + 0.533·45-s + 1.50·47-s − 0.307·49-s + 0.774·53-s − 0.0438·55-s + 0.848·57-s − 1.18·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 8.18T + 27T^{2} \) |
| 5 | \( 1 - 4.03T + 125T^{2} \) |
| 7 | \( 1 + 15.4T + 343T^{2} \) |
| 11 | \( 1 + 4.43T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.5T + 2.19e3T^{2} \) |
| 19 | \( 1 - 44.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 304.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 269.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 152.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 486.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 298.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 535.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 59.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 634.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 822.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 821.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 233.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 404.T + 9.12e5T^{2} \) |
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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
−8.385753054746654836526841403497, −7.39960601015891709316091406579, −7.12980667456003105981444609782, −5.91621818186239507492173705495, −5.06009789411099222363805913899, −3.80437227628286499269078966472, −3.29539321349210079632865578205, −2.41838772540405174453145226766, −1.61395233154655513064037651704, 0,
1.61395233154655513064037651704, 2.41838772540405174453145226766, 3.29539321349210079632865578205, 3.80437227628286499269078966472, 5.06009789411099222363805913899, 5.91621818186239507492173705495, 7.12980667456003105981444609782, 7.39960601015891709316091406579, 8.385753054746654836526841403497
