| L(s) = 1 | + 4·5-s − 6.92i·7-s − 6.92i·11-s + 18·17-s + 20.7i·19-s + 41.5i·23-s − 9·25-s − 4·29-s + 48.4i·31-s − 27.7i·35-s + 72·37-s + 18·41-s + 62.3i·43-s − 41.5i·47-s + 1.00·49-s + ⋯ |
| L(s) = 1 | + 0.800·5-s − 0.989i·7-s − 0.629i·11-s + 1.05·17-s + 1.09i·19-s + 1.80i·23-s − 0.359·25-s − 0.137·29-s + 1.56i·31-s − 0.791i·35-s + 1.94·37-s + 0.439·41-s + 1.45i·43-s − 0.884i·47-s + 0.0204·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.566100895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.566100895\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4T + 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 + 6.92iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 18T + 289T^{2} \) |
| 19 | \( 1 - 20.7iT - 361T^{2} \) |
| 23 | \( 1 - 41.5iT - 529T^{2} \) |
| 29 | \( 1 + 4T + 841T^{2} \) |
| 31 | \( 1 - 48.4iT - 961T^{2} \) |
| 37 | \( 1 - 72T + 1.36e3T^{2} \) |
| 41 | \( 1 - 18T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 41.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 44T + 2.80e3T^{2} \) |
| 59 | \( 1 + 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 72T + 3.72e3T^{2} \) |
| 67 | \( 1 - 20.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 41.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 62.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 131. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 126T + 7.92e3T^{2} \) |
| 97 | \( 1 - 110T + 9.40e3T^{2} \) |
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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
−8.922676147648971169630142531606, −7.78619995681552466229885532373, −7.54661473075929990952076346392, −6.29821566146144892784002987822, −5.80813606062610157076197419598, −4.94635786834031838701630724880, −3.78339677627036024396135804544, −3.18447255239663923278878685409, −1.75156446775331973202373380216, −0.941770219425696980968359119396,
0.78262324849015727884478115642, 2.28962900927643314952042269868, 2.57153676511643879580590706307, 4.06410165079592737234686610873, 4.93575695541660474797804156057, 5.81062172813323384564936501985, 6.26397869321960292765317028505, 7.31233820175240598183256722868, 8.083622918615198645903858220743, 9.013018884498517853558631023492
