| L(s) = 1 | + 1.61·3-s + 3.61·5-s − 7-s − 0.381·9-s − 6.09·11-s + 5.85·15-s − 17-s + 2.85·19-s − 1.61·21-s + 2·23-s + 8.09·25-s − 5.47·27-s − 8.09·29-s − 5.47·31-s − 9.85·33-s − 3.61·35-s + 8.94·37-s − 2.76·41-s − 7.09·43-s − 1.38·45-s − 3·47-s + 49-s − 1.61·51-s − 4.70·53-s − 22.0·55-s + 4.61·57-s − 9.47·59-s + ⋯ |
| L(s) = 1 | + 0.934·3-s + 1.61·5-s − 0.377·7-s − 0.127·9-s − 1.83·11-s + 1.51·15-s − 0.242·17-s + 0.654·19-s − 0.353·21-s + 0.417·23-s + 1.61·25-s − 1.05·27-s − 1.50·29-s − 0.982·31-s − 1.71·33-s − 0.611·35-s + 1.47·37-s − 0.431·41-s − 1.08·43-s − 0.206·45-s − 0.437·47-s + 0.142·49-s − 0.226·51-s − 0.646·53-s − 2.97·55-s + 0.611·57-s − 1.23·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 3.61T + 5T^{2} \) |
| 11 | \( 1 + 6.09T + 11T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 8.09T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 + 2.76T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 4.70T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 5.47T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 3.47T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 3.56T + 97T^{2} \) |
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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
−7.58113412484521511100125280469, −6.64333142870835793623114525545, −5.82283827035368104839948473153, −5.44728490804260760132908332222, −4.76553430838002997016891632023, −3.46432497588690018978431996770, −2.88713981860330029752294120556, −2.29749775324832457600724185752, −1.61112925864846247875673134891, 0,
1.61112925864846247875673134891, 2.29749775324832457600724185752, 2.88713981860330029752294120556, 3.46432497588690018978431996770, 4.76553430838002997016891632023, 5.44728490804260760132908332222, 5.82283827035368104839948473153, 6.64333142870835793623114525545, 7.58113412484521511100125280469
