L(s) = 1 | − i·3-s − i·5-s + 3.12·7-s − 9-s − 1.96·11-s − 2.52·13-s − 15-s + 3.66i·17-s − 6.78·19-s − 3.12i·21-s + (2.20 + 4.25i)23-s − 25-s + i·27-s + 2.23·29-s + 0.377i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + 1.18·7-s − 0.333·9-s − 0.593·11-s − 0.699·13-s − 0.258·15-s + 0.887i·17-s − 1.55·19-s − 0.681i·21-s + (0.460 + 0.887i)23-s − 0.200·25-s + 0.192i·27-s + 0.415·29-s + 0.0678i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.851121423\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.851121423\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 23 | \( 1 + (-2.20 - 4.25i)T \) |
good | 7 | \( 1 - 3.12T + 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 17 | \( 1 - 3.66iT - 17T^{2} \) |
| 19 | \( 1 + 6.78T + 19T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 - 0.377iT - 31T^{2} \) |
| 37 | \( 1 - 5.26iT - 37T^{2} \) |
| 41 | \( 1 - 5.40T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 1.35iT - 47T^{2} \) |
| 53 | \( 1 + 3.17iT - 53T^{2} \) |
| 59 | \( 1 + 11.9iT - 59T^{2} \) |
| 61 | \( 1 - 4.62iT - 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 15.1iT - 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 5.38T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 0.338iT - 89T^{2} \) |
| 97 | \( 1 + 11.0iT - 97T^{2} \) |
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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
−8.088221598835390615529490870570, −7.64882513429383463165181758376, −6.76669402455197227440458265825, −5.98544042827161407503158236484, −5.19137760795033495178211407191, −4.63667768702649237509941754099, −3.78654865181564492559394234936, −2.48618685271101238056656251859, −1.90121384470108843274274456962, −0.874735151153745815483948379049,
0.59679963892618232824569235581, 2.25241670779331774049407040882, 2.57213059795473921702505649464, 3.84538957709734257079114884006, 4.63168765232616812831081081253, 5.03078611542230693738173530196, 5.94406770933167578067146475057, 6.75914067470307684643769916426, 7.62219764005118262441672885411, 8.041649987942166372891152267514