L(s) = 1 | − 5-s − 2.49i·11-s − 6.80i·13-s + 7.80·17-s + 2.46i·19-s − 3.62i·23-s + 25-s + 2.55i·29-s + 4.25i·31-s + 1.85·37-s − 4.23·41-s + 8.70·43-s + 1.03·47-s + 0.0572i·53-s + 2.49i·55-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.751i·11-s − 1.88i·13-s + 1.89·17-s + 0.564i·19-s − 0.756i·23-s + 0.200·25-s + 0.473i·29-s + 0.764i·31-s + 0.304·37-s − 0.661·41-s + 1.32·43-s + 0.150·47-s + 0.00787i·53-s + 0.336i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.819521593\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.819521593\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2.49iT - 11T^{2} \) |
| 13 | \( 1 + 6.80iT - 13T^{2} \) |
| 17 | \( 1 - 7.80T + 17T^{2} \) |
| 19 | \( 1 - 2.46iT - 19T^{2} \) |
| 23 | \( 1 + 3.62iT - 23T^{2} \) |
| 29 | \( 1 - 2.55iT - 29T^{2} \) |
| 31 | \( 1 - 4.25iT - 31T^{2} \) |
| 37 | \( 1 - 1.85T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 - 1.03T + 47T^{2} \) |
| 53 | \( 1 - 0.0572iT - 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 3.10iT - 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 8.13iT - 71T^{2} \) |
| 73 | \( 1 - 4.15iT - 73T^{2} \) |
| 79 | \( 1 + 4.16T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 9.96T + 89T^{2} \) |
| 97 | \( 1 + 4.91iT - 97T^{2} \) |
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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
−7.64076442475977258937443197167, −7.09004062818193450999893815554, −5.99842152760545701950555507200, −5.59510182447658394441249934205, −4.97138369887234859416567200381, −3.84899422547430561457313273067, −3.27755301866129603066248680680, −2.71605361058766414269011815195, −1.25141266647530949161847688497, −0.51988032205579300792582433018,
1.00436066189669112297027053113, 1.93896502656517594827359541844, 2.81040610547617600728806097956, 3.90215854804411241569251055907, 4.20833017461971699762041680272, 5.15631968313107988112350033679, 5.79940381113958548414246129249, 6.74411490545187157672297677918, 7.21419249762441035304703750630, 7.80694708599009706667082401143