| L(s) = 1 | + 2-s + 4-s + 3.27·5-s + 2i·7-s + 8-s + 3.27·10-s − 4i·11-s + 3.27i·13-s + 2i·14-s + 16-s + 1.27i·17-s + 1.27i·19-s + 3.27·20-s − 4i·22-s + 5.72·25-s + 3.27i·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.46·5-s + 0.755i·7-s + 0.353·8-s + 1.03·10-s − 1.20i·11-s + 0.908i·13-s + 0.534i·14-s + 0.250·16-s + 0.309i·17-s + 0.292i·19-s + 0.732·20-s − 0.852i·22-s + 1.14·25-s + 0.642i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.91846 + 0.293483i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.91846 + 0.293483i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (-6.27 + 1.27i)T \) |
| good | 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 3.27iT - 13T^{2} \) |
| 17 | \( 1 - 1.27iT - 17T^{2} \) |
| 19 | \( 1 - 1.27iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.54iT - 29T^{2} \) |
| 31 | \( 1 + 9.27T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 43 | \( 1 - 2.54T + 43T^{2} \) |
| 47 | \( 1 - 12.5iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 5.27T + 59T^{2} \) |
| 61 | \( 1 - 8.54T + 61T^{2} \) |
| 67 | \( 1 + 9.27iT - 67T^{2} \) |
| 71 | \( 1 + 11.2iT - 71T^{2} \) |
| 73 | \( 1 + 7.27T + 73T^{2} \) |
| 79 | \( 1 + 8.54iT - 79T^{2} \) |
| 83 | \( 1 + 1.27T + 83T^{2} \) |
| 89 | \( 1 - 7.82iT - 89T^{2} \) |
| 97 | \( 1 - 14.5iT - 97T^{2} \) |
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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
−10.53632139018069701600101227683, −9.417630370540650809661896390559, −8.959410036305981340937946324163, −7.74949547923341491437516907252, −6.38815909203665574376197535034, −5.95404156460119893215692526597, −5.25202576468229547109174389445, −3.92471920061285916233415876945, −2.64922524999579859552961834831, −1.73905711092661945783425174461,
1.53705612134866705850724702106, 2.63210587861110098868197992521, 3.91784330776504608632291943277, 5.13993972863958548471760640094, 5.62442280365082061927474199644, 6.91916277560660723815704735808, 7.29575009571569128872442553866, 8.774288262750250377829794369773, 9.743127723647726074790984706497, 10.35163672087874851480022473903
