| L(s) = 1 | + 2.27·3-s + 2.27·5-s + 0.121·7-s + 2.18·9-s − 2.39·11-s + 13-s + 5.18·15-s − 0.943·17-s + 7.86·19-s + 0.277·21-s + 0.665·23-s + 0.186·25-s − 1.85·27-s + 7.22·29-s + 7.48·31-s − 5.46·33-s + 0.277·35-s − 6.52·37-s + 2.27·39-s − 2.66·41-s + 6.03·43-s + 4.97·45-s − 0.676·47-s − 6.98·49-s − 2.14·51-s + 10.2·53-s − 5.46·55-s + ⋯ |
| L(s) = 1 | + 1.31·3-s + 1.01·5-s + 0.0460·7-s + 0.728·9-s − 0.723·11-s + 0.277·13-s + 1.33·15-s − 0.228·17-s + 1.80·19-s + 0.0605·21-s + 0.138·23-s + 0.0373·25-s − 0.356·27-s + 1.34·29-s + 1.34·31-s − 0.951·33-s + 0.0468·35-s − 1.07·37-s + 0.364·39-s − 0.416·41-s + 0.920·43-s + 0.742·45-s − 0.0987·47-s − 0.997·49-s − 0.300·51-s + 1.40·53-s − 0.736·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.772210039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.772210039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 - 2.27T + 5T^{2} \) |
| 7 | \( 1 - 0.121T + 7T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 17 | \( 1 + 0.943T + 17T^{2} \) |
| 19 | \( 1 - 7.86T + 19T^{2} \) |
| 23 | \( 1 - 0.665T + 23T^{2} \) |
| 29 | \( 1 - 7.22T + 29T^{2} \) |
| 31 | \( 1 - 7.48T + 31T^{2} \) |
| 37 | \( 1 + 6.52T + 37T^{2} \) |
| 41 | \( 1 + 2.66T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 + 0.676T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 2.15T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 6.13T + 73T^{2} \) |
| 79 | \( 1 + 2.68T + 79T^{2} \) |
| 83 | \( 1 - 5.17T + 83T^{2} \) |
| 89 | \( 1 - 1.75T + 89T^{2} \) |
| 97 | \( 1 + 2.90T + 97T^{2} \) |
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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
−8.613660987067059548374081928899, −8.043932695260702551101085333637, −7.30281809938641429791253428616, −6.43461184691290099352114261703, −5.53357082058304104112654515327, −4.86614333707326781454675312705, −3.66852467331440752722089218389, −2.86920837552729298653558477345, −2.28451465009860861771645923141, −1.17721018495011318312341725790,
1.17721018495011318312341725790, 2.28451465009860861771645923141, 2.86920837552729298653558477345, 3.66852467331440752722089218389, 4.86614333707326781454675312705, 5.53357082058304104112654515327, 6.43461184691290099352114261703, 7.30281809938641429791253428616, 8.043932695260702551101085333637, 8.613660987067059548374081928899
