L(s) = 1 | − 3-s + 3.80i·5-s − 5.13i·7-s + 9-s − 0.334i·11-s − 3.80i·15-s − 4.13·17-s − 5.94i·19-s + 5.13i·21-s − 0.334·23-s − 9.47·25-s − 27-s − 0.195·29-s + 4.80i·31-s + 0.334i·33-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.70i·5-s − 1.94i·7-s + 0.333·9-s − 0.100i·11-s − 0.982i·15-s − 1.00·17-s − 1.36i·19-s + 1.12i·21-s − 0.0698·23-s − 1.89·25-s − 0.192·27-s − 0.0363·29-s + 0.862i·31-s + 0.0582i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6461162858\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6461162858\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3.80iT - 5T^{2} \) |
| 7 | \( 1 + 5.13iT - 7T^{2} \) |
| 11 | \( 1 + 0.334iT - 11T^{2} \) |
| 17 | \( 1 + 4.13T + 17T^{2} \) |
| 19 | \( 1 + 5.94iT - 19T^{2} \) |
| 23 | \( 1 + 0.334T + 23T^{2} \) |
| 29 | \( 1 + 0.195T + 29T^{2} \) |
| 31 | \( 1 - 4.80iT - 31T^{2} \) |
| 37 | \( 1 - 2.13iT - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 - 3.66iT - 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 - 6.27iT - 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 4.33iT - 71T^{2} \) |
| 73 | \( 1 - 10.6iT - 73T^{2} \) |
| 79 | \( 1 + 0.134T + 79T^{2} \) |
| 83 | \( 1 + 13.2iT - 83T^{2} \) |
| 89 | \( 1 + 0.390iT - 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−8.630744693885063423280506795545, −7.49859777647556059555662176032, −7.11599426696359438989744041845, −6.71920048847174809509943402126, −6.01294553001309350091571511308, −4.75374123472213745046646353458, −4.14192347461549325542775835977, −3.31394948679637638342325092007, −2.45415788013483481117057706988, −1.03782175870250077972096546024,
0.22370731480916464827012271222, 1.65962899894441374258369332099, 2.28328209026800020561620087986, 3.74898868184369707262172114265, 4.65961774506325335101078598254, 5.25645134628765644420866594704, 5.83369812010616674972123596822, 6.35176102961494063774794786424, 7.69962796350154028671529411194, 8.326124397606718031197466899324