| L(s) = 1 | + 2-s + 1.80·3-s + 4-s + 0.972·5-s + 1.80·6-s + 0.354·7-s + 8-s + 0.267·9-s + 0.972·10-s + 0.744·11-s + 1.80·12-s + 3.53·13-s + 0.354·14-s + 1.75·15-s + 16-s − 17-s + 0.267·18-s − 5.43·19-s + 0.972·20-s + 0.640·21-s + 0.744·22-s + 7.86·23-s + 1.80·24-s − 4.05·25-s + 3.53·26-s − 4.93·27-s + 0.354·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.04·3-s + 0.5·4-s + 0.435·5-s + 0.737·6-s + 0.133·7-s + 0.353·8-s + 0.0891·9-s + 0.307·10-s + 0.224·11-s + 0.521·12-s + 0.981·13-s + 0.0946·14-s + 0.454·15-s + 0.250·16-s − 0.242·17-s + 0.0630·18-s − 1.24·19-s + 0.217·20-s + 0.139·21-s + 0.158·22-s + 1.63·23-s + 0.368·24-s − 0.810·25-s + 0.694·26-s − 0.950·27-s + 0.0669·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.436667348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.436667348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.80T + 3T^{2} \) |
| 5 | \( 1 - 0.972T + 5T^{2} \) |
| 7 | \( 1 - 0.354T + 7T^{2} \) |
| 11 | \( 1 - 0.744T + 11T^{2} \) |
| 13 | \( 1 - 3.53T + 13T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 - 7.86T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 - 4.82T + 37T^{2} \) |
| 41 | \( 1 - 2.92T + 41T^{2} \) |
| 43 | \( 1 + 2.89T + 43T^{2} \) |
| 47 | \( 1 - 2.12T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 - 5.69T + 71T^{2} \) |
| 73 | \( 1 - 7.54T + 73T^{2} \) |
| 79 | \( 1 - 0.541T + 79T^{2} \) |
| 83 | \( 1 + 6.49T + 83T^{2} \) |
| 89 | \( 1 - 0.320T + 89T^{2} \) |
| 97 | \( 1 + 3.05T + 97T^{2} \) |
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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
−9.000852461407020000376582870882, −8.397256241302741227157703182630, −7.74564070744080817091719737647, −6.46663100350229372496382119996, −6.22190705064896330624564761634, −4.93453887829875664379094434960, −4.19543767649878720885613980462, −3.17020831222083462078076056987, −2.53101955512265964822346640370, −1.39799250899288334988781324184,
1.39799250899288334988781324184, 2.53101955512265964822346640370, 3.17020831222083462078076056987, 4.19543767649878720885613980462, 4.93453887829875664379094434960, 6.22190705064896330624564761634, 6.46663100350229372496382119996, 7.74564070744080817091719737647, 8.397256241302741227157703182630, 9.000852461407020000376582870882
