L(s) = 1 | − 1.59·5-s − 1.16i·7-s − 1.98i·11-s − 0.164i·13-s − 3.57i·17-s − 3.16·19-s + 1.21·23-s − 2.44·25-s − 4.98·29-s + 6.64i·31-s + 1.85i·35-s + 1.10i·37-s − 2.02i·41-s − 8.34·43-s − 2.21·47-s + ⋯ |
L(s) = 1 | − 0.714·5-s − 0.440i·7-s − 0.597i·11-s − 0.0455i·13-s − 0.867i·17-s − 0.725·19-s + 0.253·23-s − 0.489·25-s − 0.925·29-s + 1.19i·31-s + 0.314i·35-s + 0.181i·37-s − 0.316i·41-s − 1.27·43-s − 0.323·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4956820987\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4956820987\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 + 1.16iT - 7T^{2} \) |
| 11 | \( 1 + 1.98iT - 11T^{2} \) |
| 13 | \( 1 + 0.164iT - 13T^{2} \) |
| 17 | \( 1 + 3.57iT - 17T^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 4.98T + 29T^{2} \) |
| 31 | \( 1 - 6.64iT - 31T^{2} \) |
| 37 | \( 1 - 1.10iT - 37T^{2} \) |
| 41 | \( 1 + 2.02iT - 41T^{2} \) |
| 43 | \( 1 + 8.34T + 43T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 - 0.888iT - 59T^{2} \) |
| 61 | \( 1 - 9.74iT - 61T^{2} \) |
| 67 | \( 1 + 3.98T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 - 0.879T + 73T^{2} \) |
| 79 | \( 1 - 2.91iT - 79T^{2} \) |
| 83 | \( 1 + 13.9iT - 83T^{2} \) |
| 89 | \( 1 + 8.41iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−8.502295999204154004773145714905, −7.57868597986250247776613051109, −7.15278593401004522611926099208, −6.34438716718033935039412884580, −5.46161410597305957670512342797, −4.72051956865906503543329133567, −3.85893420113703245315311190900, −3.29198696522850891914883275702, −2.22153904528830357219537878983, −0.948317142500891894291550594658,
0.15374456949785480882940290913, 1.70918647861149709158599739255, 2.48424028982005410012441935519, 3.69809574275917314166739803621, 4.11969171645886604736336078173, 5.07310429937096153910374858956, 5.84416219906010039316854778812, 6.60585711178122380136279308344, 7.34165444854191970993980503066, 8.086366115235920662163106503757