L(s) = 1 | − 3.27·3-s + 2.17i·5-s − i·7-s + 7.72·9-s + i·11-s + (3.04 − 1.92i)13-s − 7.13i·15-s − 4.89·17-s − 2.89i·19-s + 3.27i·21-s − 3.83·23-s + 0.253·25-s − 15.4·27-s + 3.96·29-s + 0.338i·31-s + ⋯ |
L(s) = 1 | − 1.89·3-s + 0.974i·5-s − 0.377i·7-s + 2.57·9-s + 0.301i·11-s + (0.845 − 0.533i)13-s − 1.84i·15-s − 1.18·17-s − 0.664i·19-s + 0.714i·21-s − 0.799·23-s + 0.0506·25-s − 2.97·27-s + 0.737·29-s + 0.0607i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08286184607\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08286184607\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-3.04 + 1.92i)T \) |
good | 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 - 2.17iT - 5T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 2.89iT - 19T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 - 0.338iT - 31T^{2} \) |
| 37 | \( 1 + 9.41iT - 37T^{2} \) |
| 41 | \( 1 - 1.25iT - 41T^{2} \) |
| 43 | \( 1 + 7.90T + 43T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 - 9.32iT - 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 1.95iT - 67T^{2} \) |
| 71 | \( 1 + 5.38iT - 71T^{2} \) |
| 73 | \( 1 - 0.975iT - 73T^{2} \) |
| 79 | \( 1 - 5.42T + 79T^{2} \) |
| 83 | \( 1 + 4.61iT - 83T^{2} \) |
| 89 | \( 1 - 2.06iT - 89T^{2} \) |
| 97 | \( 1 - 0.930iT - 97T^{2} \) |
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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
−7.85578275727355805011804307264, −7.07971238345531987497032609342, −6.55981015209233373890368615422, −6.11104122230072904573094955436, −5.24327492285090879806114995420, −4.46962525898277058670260452002, −3.76196057767867750227473356110, −2.46824424965573703839944385001, −1.19657908161039099101860782579, −0.03898014782673893639542785052,
1.08273211922506637129884620623, 1.93058276612862132516626153213, 3.72118080225812792018363855300, 4.49560496365340165975749066547, 5.09906824348899299490219135981, 5.72124515623545115327657446513, 6.53781659588358290379276910607, 6.79197851721027074042413389602, 8.217327958090179661062929059339, 8.594743884408823637889962398276