| L(s) = 1 | + 3.49·5-s + 4.74·7-s + 2.44·11-s + 1.24·13-s − 17-s − 0.444·19-s − 0.188·23-s + 7.23·25-s − 4.84·29-s − 2.84·31-s + 16.5·35-s + 4.84·37-s − 0.865·41-s − 0.552·43-s + 6.99·47-s + 15.4·49-s + 4.10·53-s + 8.55·55-s − 0.107·59-s − 13.9·61-s + 4.34·65-s + 13.2·67-s − 12.3·71-s − 9.59·73-s + 11.5·77-s + 6.25·79-s + 11.4·83-s + ⋯ |
| L(s) = 1 | + 1.56·5-s + 1.79·7-s + 0.737·11-s + 0.344·13-s − 0.242·17-s − 0.101·19-s − 0.0393·23-s + 1.44·25-s − 0.900·29-s − 0.511·31-s + 2.80·35-s + 0.797·37-s − 0.135·41-s − 0.0842·43-s + 1.02·47-s + 2.21·49-s + 0.564·53-s + 1.15·55-s − 0.0140·59-s − 1.78·61-s + 0.539·65-s + 1.61·67-s − 1.46·71-s − 1.12·73-s + 1.32·77-s + 0.703·79-s + 1.26·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.721495814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.721495814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 3.49T + 5T^{2} \) |
| 7 | \( 1 - 4.74T + 7T^{2} \) |
| 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 19 | \( 1 + 0.444T + 19T^{2} \) |
| 23 | \( 1 + 0.188T + 23T^{2} \) |
| 29 | \( 1 + 4.84T + 29T^{2} \) |
| 31 | \( 1 + 2.84T + 31T^{2} \) |
| 37 | \( 1 - 4.84T + 37T^{2} \) |
| 41 | \( 1 + 0.865T + 41T^{2} \) |
| 43 | \( 1 + 0.552T + 43T^{2} \) |
| 47 | \( 1 - 6.99T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 + 0.107T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 9.59T + 73T^{2} \) |
| 79 | \( 1 - 6.25T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 6.88T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.320274517356335129833612445491, −7.58527067421446456363331744990, −6.76041808518990231640365636533, −5.91608129144980154287633744716, −5.45477784063467623195085876210, −4.66562139208357785726653284808, −3.90711940790849831771358496941, −2.54503411858960073909887813240, −1.79373253590848337619221748681, −1.22845013377888238466148035529,
1.22845013377888238466148035529, 1.79373253590848337619221748681, 2.54503411858960073909887813240, 3.90711940790849831771358496941, 4.66562139208357785726653284808, 5.45477784063467623195085876210, 5.91608129144980154287633744716, 6.76041808518990231640365636533, 7.58527067421446456363331744990, 8.320274517356335129833612445491
