L(s) = 1 | − 0.477i·3-s + (1.61 − 2.09i)7-s + 2.77·9-s − 4.66i·11-s + 3.77·13-s + 3.77·17-s + 7.42·19-s + (−1 − 0.772i)21-s − 3.23·23-s − 2.75i·27-s − 3.77·29-s + 0.954·31-s − 2.22·33-s + 5.54i·37-s − 1.80i·39-s + ⋯ |
L(s) = 1 | − 0.275i·3-s + (0.611 − 0.791i)7-s + 0.924·9-s − 1.40i·11-s + 1.04·13-s + 0.914·17-s + 1.70·19-s + (−0.218 − 0.168i)21-s − 0.674·23-s − 0.530i·27-s − 0.700·29-s + 0.171·31-s − 0.387·33-s + 0.911i·37-s − 0.288i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.429182005\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.429182005\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.61 + 2.09i)T \) |
good | 3 | \( 1 + 0.477iT - 3T^{2} \) |
| 11 | \( 1 + 4.66iT - 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 - 3.77T + 17T^{2} \) |
| 19 | \( 1 - 7.42T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 3.77T + 29T^{2} \) |
| 31 | \( 1 - 0.954T + 31T^{2} \) |
| 37 | \( 1 - 5.54iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 7.89iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 9.33T + 59T^{2} \) |
| 61 | \( 1 + 13.5iT - 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 - 9.33iT - 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 11.1iT - 79T^{2} \) |
| 83 | \( 1 + 12.5iT - 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 8.22T + 97T^{2} \) |
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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
−8.346919546739642688642656060594, −7.999098229845931692187138417005, −7.25167541683716772052778059956, −6.39689216803090802982088749910, −5.63275437117388663208409495889, −4.76922114952320805672987785787, −3.71527444953843132440031513319, −3.20995974758740684941902732128, −1.50077407212598340277814802791, −0.955213143502220738569083611089,
1.33932167908274136178795909338, 2.10013045151241407310677695078, 3.45495558540525798168098791130, 4.16238078070481412328387990734, 5.21759305213080914774430980029, 5.55781766025061499831766663102, 6.82420281828039865834341780136, 7.42391537522051765383497500449, 8.157439555323074687813680979230, 8.989251665505878324533227976150