| L(s) = 1 | + i·3-s − 3i·7-s − 9-s + 2·11-s − 3i·13-s − 6i·17-s + 7·19-s + 3·21-s + 6i·23-s − i·27-s + 2·29-s − 5·31-s + 2i·33-s − 10i·37-s + 3·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.13i·7-s − 0.333·9-s + 0.603·11-s − 0.832i·13-s − 1.45i·17-s + 1.60·19-s + 0.654·21-s + 1.25i·23-s − 0.192i·27-s + 0.371·29-s − 0.898·31-s + 0.348i·33-s − 1.64i·37-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44010 - 0.339963i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44010 - 0.339963i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 3iT - 43T^{2} \) |
| 47 | \( 1 - 10iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−10.67371455071584097676060671901, −9.512403782821587488168576259098, −9.311727942086990365158212016738, −7.59654940602478680320763469857, −7.39498139352411021883189878660, −5.89986252985925105185926952266, −4.98656065022706237609613720348, −3.90616489776559888802834786731, −3.00828295337730168631484694991, −0.942396002529228916396980195111,
1.52137860185798936535033025746, 2.72428241094245306777605376979, 4.07664135888561656889986201682, 5.41116661655413674771725039813, 6.23262787886658877180646550497, 7.05756624392090323547003701269, 8.234180655844527737670026617350, 8.870499529610187065756594702120, 9.706034971630114849990642581933, 10.86017073575357971655165105441
