| L(s) = 1 | + 1.50·2-s + 3-s + 0.251·4-s + 3.44·5-s + 1.50·6-s − 7-s − 2.62·8-s + 9-s + 5.16·10-s − 2.15·11-s + 0.251·12-s − 5.70·13-s − 1.50·14-s + 3.44·15-s − 4.43·16-s + 3.02·17-s + 1.50·18-s − 2.45·19-s + 0.866·20-s − 21-s − 3.22·22-s − 6.49·23-s − 2.62·24-s + 6.84·25-s − 8.56·26-s + 27-s − 0.251·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 0.125·4-s + 1.53·5-s + 0.612·6-s − 0.377·7-s − 0.927·8-s + 0.333·9-s + 1.63·10-s − 0.648·11-s + 0.0726·12-s − 1.58·13-s − 0.401·14-s + 0.888·15-s − 1.10·16-s + 0.733·17-s + 0.353·18-s − 0.563·19-s + 0.193·20-s − 0.218·21-s − 0.688·22-s − 1.35·23-s − 0.535·24-s + 1.36·25-s − 1.67·26-s + 0.192·27-s − 0.0475·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 5 | \( 1 - 3.44T + 5T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 + 5.70T + 13T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 + 2.45T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 8.55T + 29T^{2} \) |
| 31 | \( 1 - 5.50T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 + 9.47T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 - 8.10T + 73T^{2} \) |
| 79 | \( 1 + 6.63T + 79T^{2} \) |
| 83 | \( 1 - 9.06T + 83T^{2} \) |
| 89 | \( 1 + 0.981T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−7.50751440144686475175924568613, −6.38299199589657573731168416155, −6.06177926261746917284034836229, −5.31225237374643722887978981131, −4.75884556200995070219242717445, −3.97534454870223542854266681299, −2.93549941757301741027222722151, −2.53373512928238270189898096304, −1.74633835790132302921965789063, 0,
1.74633835790132302921965789063, 2.53373512928238270189898096304, 2.93549941757301741027222722151, 3.97534454870223542854266681299, 4.75884556200995070219242717445, 5.31225237374643722887978981131, 6.06177926261746917284034836229, 6.38299199589657573731168416155, 7.50751440144686475175924568613
