| L(s) = 1 | + 2·3-s + 16·5-s − 23·9-s + 24·11-s + 68·13-s + 32·15-s − 54·17-s + 46·19-s + 176·23-s + 131·25-s − 100·27-s − 174·29-s + 116·31-s + 48·33-s + 74·37-s + 136·39-s + 10·41-s − 480·43-s − 368·45-s + 572·47-s − 108·51-s − 162·53-s + 384·55-s + 92·57-s + 86·59-s + 904·61-s + 1.08e3·65-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.43·5-s − 0.851·9-s + 0.657·11-s + 1.45·13-s + 0.550·15-s − 0.770·17-s + 0.555·19-s + 1.59·23-s + 1.04·25-s − 0.712·27-s − 1.11·29-s + 0.672·31-s + 0.253·33-s + 0.328·37-s + 0.558·39-s + 0.0380·41-s − 1.70·43-s − 1.21·45-s + 1.77·47-s − 0.296·51-s − 0.419·53-s + 0.941·55-s + 0.213·57-s + 0.189·59-s + 1.89·61-s + 2.07·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.937362874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.937362874\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 - 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 46 T + p^{3} T^{2} \) |
| 23 | \( 1 - 176 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 116 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 - 10 T + p^{3} T^{2} \) |
| 43 | \( 1 + 480 T + p^{3} T^{2} \) |
| 47 | \( 1 - 572 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 86 T + p^{3} T^{2} \) |
| 61 | \( 1 - 904 T + p^{3} T^{2} \) |
| 67 | \( 1 - 660 T + p^{3} T^{2} \) |
| 71 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 73 | \( 1 + 770 T + p^{3} T^{2} \) |
| 79 | \( 1 + 904 T + p^{3} T^{2} \) |
| 83 | \( 1 + 682 T + p^{3} T^{2} \) |
| 89 | \( 1 - 102 T + p^{3} T^{2} \) |
| 97 | \( 1 - 218 T + p^{3} T^{2} \) |
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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
−10.95649948767904685507391681344, −9.780629161485940896165112396476, −9.023438555707756129960017709377, −8.468690323536810210241469157040, −6.91290056387359129743431381542, −6.06672867706294624725874693725, −5.25741890527043516051349804130, −3.66261309830601877684713893535, −2.47579550670012325764007220086, −1.23254297268439354173119295852,
1.23254297268439354173119295852, 2.47579550670012325764007220086, 3.66261309830601877684713893535, 5.25741890527043516051349804130, 6.06672867706294624725874693725, 6.91290056387359129743431381542, 8.468690323536810210241469157040, 9.023438555707756129960017709377, 9.780629161485940896165112396476, 10.95649948767904685507391681344
