| L(s) = 1 | − 3·3-s + 9.12·5-s + 7·7-s + 9·9-s − 34.3·11-s + 20.2·13-s − 27.3·15-s − 120.·17-s + 127.·19-s − 21·21-s + 95.0·23-s − 41.7·25-s − 27·27-s − 113.·29-s + 126.·31-s + 103.·33-s + 63.8·35-s − 201.·37-s − 60.7·39-s − 444.·41-s − 141.·43-s + 82.1·45-s − 591.·47-s + 49·49-s + 361.·51-s − 5.78·53-s − 313.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.815·5-s + 0.377·7-s + 0.333·9-s − 0.941·11-s + 0.431·13-s − 0.471·15-s − 1.71·17-s + 1.54·19-s − 0.218·21-s + 0.861·23-s − 0.334·25-s − 0.192·27-s − 0.729·29-s + 0.732·31-s + 0.543·33-s + 0.308·35-s − 0.896·37-s − 0.249·39-s − 1.69·41-s − 0.501·43-s + 0.271·45-s − 1.83·47-s + 0.142·49-s + 0.991·51-s − 0.0149·53-s − 0.768·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 - 9.12T + 125T^{2} \) |
| 11 | \( 1 + 34.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 120.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 95.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 113.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 126.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 201.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 591.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 5.78T + 1.48e5T^{2} \) |
| 59 | \( 1 - 348.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 532.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 661.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 324.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 613.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 643.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 908.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 662.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 221.T + 9.12e5T^{2} \) |
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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
−8.878984040629501092904587944253, −8.059520653866559443269917575828, −7.01900024097498746780488423647, −6.38027913229912710493952305807, −5.22842425697354228994723542133, −5.03476236761721435250806358189, −3.59720703001990037068036980938, −2.37349443662257919856353797045, −1.39643720064341018433214824612, 0,
1.39643720064341018433214824612, 2.37349443662257919856353797045, 3.59720703001990037068036980938, 5.03476236761721435250806358189, 5.22842425697354228994723542133, 6.38027913229912710493952305807, 7.01900024097498746780488423647, 8.059520653866559443269917575828, 8.878984040629501092904587944253
