| L(s) = 1 | + (−1 − i)2-s + i·3-s + 2i·4-s − 2i·5-s + (1 − i)6-s + (2 − 2i)8-s − 9-s + (−2 + 2i)10-s − 2·12-s + 4i·13-s + 2·15-s − 4·16-s + 2·17-s + (1 + i)18-s − 4i·19-s + 4·20-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + 0.577i·3-s + i·4-s − 0.894i·5-s + (0.408 − 0.408i)6-s + (0.707 − 0.707i)8-s − 0.333·9-s + (−0.632 + 0.632i)10-s − 0.577·12-s + 1.10i·13-s + 0.516·15-s − 16-s + 0.485·17-s + (0.235 + 0.235i)18-s − 0.917i·19-s + 0.894·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.098914860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.098914860\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.446565386270717925445284835025, −9.117614241147052395197514621246, −8.393683848736485676463918425699, −7.43915974760815993445138122382, −6.45513664356533262568768238973, −5.03748126292551446423443987385, −4.41548226215418089336467678522, −3.39010907170023209291547720816, −2.17892342853626450852854346549, −0.813668385383590047815236074732,
0.987937285693809796641341269945, 2.40227507722142369851997851184, 3.53936237947993429265750998115, 5.20907734492391596193513300061, 5.83832152770182513845811320813, 6.79985626069541508465782520373, 7.36815871921938257654508383648, 8.071909124763718465583624476350, 8.896127779672511729205001468780, 9.845002913129130323451816169974
