L(s) = 1 | + 3.13i·3-s + 3.91i·5-s − 2.98i·7-s − 6.81·9-s + 5.81i·11-s + 1.72·13-s − 12.2·15-s + (−2.58 − 3.21i)17-s − 0.658·19-s + 9.34·21-s − 9.22i·23-s − 10.3·25-s − 11.9i·27-s − 10.2i·29-s + 5.64i·31-s + ⋯ |
L(s) = 1 | + 1.80i·3-s + 1.75i·5-s − 1.12i·7-s − 2.27·9-s + 1.75i·11-s + 0.477·13-s − 3.16·15-s + (−0.626 − 0.779i)17-s − 0.151·19-s + 2.03·21-s − 1.92i·23-s − 2.06·25-s − 2.29i·27-s − 1.89i·29-s + 1.01i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1866376737\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1866376737\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + (2.58 + 3.21i)T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 3.13iT - 3T^{2} \) |
| 5 | \( 1 - 3.91iT - 5T^{2} \) |
| 7 | \( 1 + 2.98iT - 7T^{2} \) |
| 11 | \( 1 - 5.81iT - 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 19 | \( 1 + 0.658T + 19T^{2} \) |
| 23 | \( 1 + 9.22iT - 23T^{2} \) |
| 29 | \( 1 + 10.2iT - 29T^{2} \) |
| 31 | \( 1 - 5.64iT - 31T^{2} \) |
| 37 | \( 1 - 3.51iT - 37T^{2} \) |
| 41 | \( 1 + 0.336iT - 41T^{2} \) |
| 43 | \( 1 + 2.85T + 43T^{2} \) |
| 47 | \( 1 + 4.71T + 47T^{2} \) |
| 53 | \( 1 + 3.06T + 53T^{2} \) |
| 61 | \( 1 + 4.36iT - 61T^{2} \) |
| 67 | \( 1 - 4.44T + 67T^{2} \) |
| 71 | \( 1 + 10.8iT - 71T^{2} \) |
| 73 | \( 1 - 3.72iT - 73T^{2} \) |
| 79 | \( 1 + 5.49iT - 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 9.45T + 89T^{2} \) |
| 97 | \( 1 - 9.73iT - 97T^{2} \) |
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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
−9.445579652295108878490919389112, −8.393421311301739248893530760012, −7.53988500336908293427027424246, −6.71348645249315150857904490588, −6.37116160741060816258630065159, −5.00378381320817764089953227544, −4.36677525439476819730649059288, −3.92028791018785538504147536593, −2.97540378225554353888262065050, −2.29003341511536415118518949076,
0.05310055410094804374390805282, 1.25279261467420785202388675120, 1.67230273496818074355762029901, 2.85328328529584026544973996980, 3.84602935094540226736159641760, 5.32687638121387707511874996646, 5.64880962120692947468419047548, 6.14361054127729133612364066042, 7.13979246809194214028084829712, 8.115471097393173461100548459069