| L(s) = 1 | + (−1 + i)2-s − i·3-s − 2i·4-s + (1 + i)6-s − 2·7-s + (2 + 2i)8-s − 9-s + 4i·11-s − 2·12-s + (2 − 2i)14-s − 4·16-s + 6·17-s + (1 − i)18-s + 4i·19-s + 2i·21-s + (−4 − 4i)22-s + ⋯ |
| L(s) = 1 | + (−0.707 + 0.707i)2-s − 0.577i·3-s − i·4-s + (0.408 + 0.408i)6-s − 0.755·7-s + (0.707 + 0.707i)8-s − 0.333·9-s + 1.20i·11-s − 0.577·12-s + (0.534 − 0.534i)14-s − 16-s + 1.45·17-s + (0.235 − 0.235i)18-s + 0.917i·19-s + 0.436i·21-s + (−0.852 − 0.852i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.858139 + 0.355453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.858139 + 0.355453i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.34519256248327191045397691680, −9.861859387777574106379461685309, −9.026677707446116323796825846477, −7.83727342278446855732740151019, −7.41860431444945416935591347114, −6.36429654460564020163414918027, −5.69250384819205677507488475638, −4.38171239632414891983878406977, −2.66475744322071252239201683237, −1.15080276500926866166428846606,
0.824933504831208538900601064344, 2.91095070327648527113839263551, 3.41369744806173821743501061303, 4.78342985241194947751533208569, 6.05334046248431055581744507227, 7.15133876144596789457034576020, 8.263917831551586856894272402927, 8.970076493502703952046791255288, 9.760757295120013946296302677340, 10.46982852197989060389221127174
