| L(s) = 1 | + (−1 − i)2-s + 3-s + 2i·4-s + (−1 − i)6-s − 2i·7-s + (2 − 2i)8-s + 9-s + 2i·12-s + 4·13-s + (−2 + 2i)14-s − 4·16-s − 2i·17-s + (−1 − i)18-s + 4i·19-s − 2i·21-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + 0.577·3-s + i·4-s + (−0.408 − 0.408i)6-s − 0.755i·7-s + (0.707 − 0.707i)8-s + 0.333·9-s + 0.577i·12-s + 1.10·13-s + (−0.534 + 0.534i)14-s − 16-s − 0.485i·17-s + (−0.235 − 0.235i)18-s + 0.917i·19-s − 0.436i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05482 - 0.760271i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05482 - 0.760271i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.37572245916199996434400696749, −9.731919379183810237879646939798, −8.751767550179133193101784773870, −8.060124129098374446779750896102, −7.27223606196994065841182935122, −6.16726686486920548320386403878, −4.37705712725384056723531666911, −3.64955404740536252393515620589, −2.43747873830853088467340306986, −1.00918287557639663368206734040,
1.43896897148633016040909935147, 2.87443015071539085412668990651, 4.40202095210629136222690685689, 5.62975471066998538591265394359, 6.39758780459616600950820717749, 7.46693574337243420597175725925, 8.300007473630957134859413666282, 9.021310181701036461674528670792, 9.565007096933867155428190823519, 10.75558539681191237982733502033
