| L(s) = 1 | + 2i·7-s + 4·11-s − 6i·13-s + 2i·17-s − 8·19-s + 6i·23-s − 2·29-s + 4·31-s − 2i·37-s + 10·41-s − 2i·43-s − 2i·47-s + 3·49-s − 2i·53-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s + 1.20·11-s − 1.66i·13-s + 0.485i·17-s − 1.83·19-s + 1.25i·23-s − 0.371·29-s + 0.718·31-s − 0.328i·37-s + 1.56·41-s − 0.304i·43-s − 0.291i·47-s + 0.428·49-s − 0.274i·53-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.990681472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.990681472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.185989117148983439655265848516, −7.26759385787398314979198485766, −6.48584645820442027954098910832, −5.82091162209081564882910286702, −5.38280733985880451136705037685, −4.24806437155746553539322930066, −3.70308177914099961039803022630, −2.72952836326600270498315899350, −1.92925336763468307235691418587, −0.804191529473431747094285990407,
0.64922751759054705806245418579, 1.73985117507613662361081016545, 2.51328224577348842782040932230, 3.76720496821210600159941164953, 4.33367976419795829063401444198, 4.66702181980443557222511623626, 6.08669307083478761465946807948, 6.55858142983195531957121901233, 6.95438309295676997830431919735, 7.86340149941843290348118159285
