| L(s) = 1 | − 2.56i·3-s − 1.56i·7-s − 3.56·9-s − 2·11-s + 0.561i·13-s − 5.56i·17-s + 2·19-s − 4·21-s − i·23-s + 1.43i·27-s − 0.123·29-s − 8.12·31-s + 5.12i·33-s + 3.56i·37-s + 1.43·39-s + ⋯ |
| L(s) = 1 | − 1.47i·3-s − 0.590i·7-s − 1.18·9-s − 0.603·11-s + 0.155i·13-s − 1.34i·17-s + 0.458·19-s − 0.872·21-s − 0.208i·23-s + 0.276i·27-s − 0.0228·29-s − 1.45·31-s + 0.891i·33-s + 0.585i·37-s + 0.230·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7031895528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7031895528\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 2.56iT - 3T^{2} \) |
| 7 | \( 1 + 1.56iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.561iT - 13T^{2} \) |
| 17 | \( 1 + 5.56iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 + 0.123T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 - 3.56iT - 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 3.68iT - 47T^{2} \) |
| 53 | \( 1 - 4.43iT - 53T^{2} \) |
| 59 | \( 1 - 5.56T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 11.5iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 3.43iT - 73T^{2} \) |
| 79 | \( 1 - 9.12T + 79T^{2} \) |
| 83 | \( 1 + 4.68iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 3.12iT - 97T^{2} \) |
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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
−7.47838890155961083796073801930, −7.27545473404652294861636533874, −6.66396033374854039637512490753, −5.67654031345287971081739404607, −5.07668122909022947842282295778, −3.96760058852112483535357257710, −2.93696195661227507510458966370, −2.15712172700942901618379956508, −1.16331532708904731999830937027, −0.19772924119624048150674476247,
1.71035421538282281554453408446, 2.85934757836434346491820283564, 3.59834244017124888450160823220, 4.28865254756555923735584879691, 5.18618767820284179086792746395, 5.60196377084548723833188083456, 6.42219775671610489857495560110, 7.56466316933632549313297137557, 8.212681266367711477622914565193, 8.995560356410254759512991878820
