L(s) = 1 | − 221. i·5-s − 907.·7-s − 1.94e4i·11-s − 1.91e4·13-s + 1.34e5i·17-s − 1.94e5·19-s − 4.26e4i·23-s + 3.41e5·25-s − 8.14e5i·29-s − 2.03e5·31-s + 2.01e5i·35-s + 1.43e6·37-s + 2.77e6i·41-s − 5.04e6·43-s + 5.42e6i·47-s + ⋯ |
L(s) = 1 | − 0.354i·5-s − 0.377·7-s − 1.33i·11-s − 0.670·13-s + 1.61i·17-s − 1.49·19-s − 0.152i·23-s + 0.874·25-s − 1.15i·29-s − 0.220·31-s + 0.134i·35-s + 0.767·37-s + 0.982i·41-s − 1.47·43-s + 1.11i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.164226721\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.164226721\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 907.T \) |
good | 5 | \( 1 + 221. iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 1.94e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 1.91e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.34e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.94e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 4.26e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.14e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 2.03e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.43e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.77e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.04e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 5.42e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.34e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.38e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.26e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.91e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.02e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 5.42e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.10e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 1.30e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.78e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 6.89e7T + 7.83e15T^{2} \) |
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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
−10.73485417944268582190554060694, −9.861917799796366674053787550501, −8.632714502852589963653570805360, −8.134416689501160290651561396128, −6.58891609525459027356624057944, −5.89050596380712173952628689250, −4.56284339602248043655386815970, −3.48591491073338323797553644457, −2.20496528375794865316479348607, −0.791819787607958285428916898277,
0.32248595840245259041029435080, 1.97574438450403793055656299897, 2.93974471251272213645087752246, 4.36102005646089936906072491926, 5.27748473128938841543435203338, 6.85327619140510110511055106532, 7.16698916871706458926257136759, 8.625880128757723326572145219425, 9.634985095532114212468963807254, 10.33391383868484234973006511339