| L(s) = 1 | − 1.73i·5-s + 4.58i·11-s − 3.46i·13-s − 5.19i·17-s − 2.64·19-s − 4.58i·23-s + 2.00·25-s + 2.64·31-s + 7·37-s − 3.46i·41-s + 9.16i·43-s − 7.93·47-s − 3·53-s + 7.93·55-s − 7.93·59-s + ⋯ |
| L(s) = 1 | − 0.774i·5-s + 1.38i·11-s − 0.960i·13-s − 1.26i·17-s − 0.606·19-s − 0.955i·23-s + 0.400·25-s + 0.475·31-s + 1.15·37-s − 0.541i·41-s + 1.39i·43-s − 1.15·47-s − 0.412·53-s + 1.07·55-s − 1.03·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155734001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155734001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 4.58iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 2.64T + 19T^{2} \) |
| 23 | \( 1 + 4.58iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 9.16iT - 43T^{2} \) |
| 47 | \( 1 + 7.93T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + 7.93T + 59T^{2} \) |
| 61 | \( 1 + 1.73iT - 61T^{2} \) |
| 67 | \( 1 - 4.58iT - 67T^{2} \) |
| 71 | \( 1 + 9.16iT - 71T^{2} \) |
| 73 | \( 1 + 5.19iT - 73T^{2} \) |
| 79 | \( 1 - 4.58iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 1.73iT - 89T^{2} \) |
| 97 | \( 1 - 3.46iT - 97T^{2} \) |
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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
−7.82622688651431282929362283268, −6.96258571709457255452035793094, −6.35142007853778776971932263657, −5.40410275202455577239138592310, −4.63610455247356215551677235085, −4.46561033357428305403972335795, −3.10821901762098036051915793926, −2.42002007039666215648604019798, −1.33651556377193452369080395875, −0.29296936738202969112214523879,
1.21833917302875431270545295753, 2.21463308052004994395259050842, 3.15349020679841244817753895901, 3.75023798995059636919243208253, 4.54701490162146204077787482461, 5.57236973677951228243736148931, 6.27869548967833099317854665650, 6.60513063655698130857310098464, 7.53162406937374852413180753150, 8.243960494840587730429718890284
