| L(s) = 1 | − 2.78·3-s − 5·5-s − 24.1·7-s − 19.2·9-s + 67.6·11-s − 28.9·13-s + 13.9·15-s − 42.0·17-s + 109.·19-s + 67.3·21-s − 23·23-s + 25·25-s + 128.·27-s − 220.·29-s − 130.·31-s − 188.·33-s + 120.·35-s + 223.·37-s + 80.6·39-s + 361.·41-s + 354.·43-s + 96.2·45-s − 313.·47-s + 242.·49-s + 117.·51-s + 323.·53-s − 338.·55-s + ⋯ |
| L(s) = 1 | − 0.535·3-s − 0.447·5-s − 1.30·7-s − 0.712·9-s + 1.85·11-s − 0.617·13-s + 0.239·15-s − 0.600·17-s + 1.32·19-s + 0.700·21-s − 0.208·23-s + 0.200·25-s + 0.917·27-s − 1.41·29-s − 0.753·31-s − 0.994·33-s + 0.584·35-s + 0.994·37-s + 0.331·39-s + 1.37·41-s + 1.25·43-s + 0.318·45-s − 0.972·47-s + 0.707·49-s + 0.321·51-s + 0.837·53-s − 0.829·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 2.78T + 27T^{2} \) |
| 7 | \( 1 + 24.1T + 343T^{2} \) |
| 11 | \( 1 - 67.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 130.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 323.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 772.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 350.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 254.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 297.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 187.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 690.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 48.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + 267.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 602.T + 9.12e5T^{2} \) |
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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
−8.703564856985091977906062727059, −7.47141700651799492641967516617, −6.86908971872111197616881407454, −6.10273359284602262961185367918, −5.46039513184575811468507215643, −4.18009547961842036849545186850, −3.55671015467560616689016475315, −2.53824597120655175506710748741, −0.985991722290076937342190312917, 0,
0.985991722290076937342190312917, 2.53824597120655175506710748741, 3.55671015467560616689016475315, 4.18009547961842036849545186850, 5.46039513184575811468507215643, 6.10273359284602262961185367918, 6.86908971872111197616881407454, 7.47141700651799492641967516617, 8.703564856985091977906062727059
