| L(s) = 1 | + 4.67·3-s + 9.22i·5-s + 9.80i·7-s + 12.8·9-s + 11.6·11-s + 9.91i·13-s + 43.0i·15-s + 25.8·17-s − 4.35·19-s + 45.7i·21-s − 35.6i·23-s − 60.1·25-s + 17.7·27-s − 10.0i·29-s − 46.7i·31-s + ⋯ |
| L(s) = 1 | + 1.55·3-s + 1.84i·5-s + 1.40i·7-s + 1.42·9-s + 1.06·11-s + 0.762i·13-s + 2.87i·15-s + 1.51·17-s − 0.229·19-s + 2.18i·21-s − 1.55i·23-s − 2.40·25-s + 0.659·27-s − 0.347i·29-s − 1.50i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.801372139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.801372139\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35T \) |
| good | 3 | \( 1 - 4.67T + 9T^{2} \) |
| 5 | \( 1 - 9.22iT - 25T^{2} \) |
| 7 | \( 1 - 9.80iT - 49T^{2} \) |
| 11 | \( 1 - 11.6T + 121T^{2} \) |
| 13 | \( 1 - 9.91iT - 169T^{2} \) |
| 17 | \( 1 - 25.8T + 289T^{2} \) |
| 23 | \( 1 + 35.6iT - 529T^{2} \) |
| 29 | \( 1 + 10.0iT - 841T^{2} \) |
| 31 | \( 1 + 46.7iT - 961T^{2} \) |
| 37 | \( 1 - 32.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 33.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 11.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 24.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 75.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 5.42T + 3.48e3T^{2} \) |
| 61 | \( 1 - 25.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 41.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 107. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 79.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 81.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 112.T + 9.40e3T^{2} \) |
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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
−9.710910297513426640425765199504, −8.943597031630098602756268232839, −8.220516511154263261541497547334, −7.40841263459503163218390516698, −6.54502824204743176769008667351, −5.90516251971607419311200221306, −4.18592787028554163380228415675, −3.33192007206615009526245663415, −2.60761869936341952396220803876, −1.97479534375185542680585492521,
1.01122387540712451887344742729, 1.50938343206281713772081059378, 3.34385361946774762159553915485, 3.86897697943070738367311767540, 4.76331182962500754031591111374, 5.79636429439969132245569815664, 7.38260999968719448635646929114, 7.72456883258199575551859577058, 8.551467148838489858655477706003, 9.235564513242107176938650772435
