| L(s) = 1 | + 3-s + 4.16·5-s + 7-s + 9-s + 4.71·11-s + 13-s + 4.16·15-s − 0.450·17-s + 0.103·19-s + 21-s − 4.42·23-s + 12.3·25-s + 27-s + 1.89·29-s − 2.55·31-s + 4.71·33-s + 4.16·35-s − 0.450·37-s + 39-s + 4.26·41-s − 4.42·43-s + 4.16·45-s − 8.61·47-s + 49-s − 0.450·51-s + 12.6·53-s + 19.6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.86·5-s + 0.377·7-s + 0.333·9-s + 1.42·11-s + 0.277·13-s + 1.07·15-s − 0.109·17-s + 0.0236·19-s + 0.218·21-s − 0.923·23-s + 2.46·25-s + 0.192·27-s + 0.352·29-s − 0.458·31-s + 0.821·33-s + 0.703·35-s − 0.0740·37-s + 0.160·39-s + 0.666·41-s − 0.675·43-s + 0.620·45-s − 1.25·47-s + 0.142·49-s − 0.0630·51-s + 1.74·53-s + 2.64·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.159393951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.159393951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 4.16T + 5T^{2} \) |
| 11 | \( 1 - 4.71T + 11T^{2} \) |
| 17 | \( 1 + 0.450T + 17T^{2} \) |
| 19 | \( 1 - 0.103T + 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 + 0.450T + 37T^{2} \) |
| 41 | \( 1 - 4.26T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 + 8.61T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 + 7.16T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 - 1.57T + 73T^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 - 8.93T + 83T^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 + 4.67T + 97T^{2} \) |
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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
−8.593790312985305814131505869791, −7.69203486300594974136712553247, −6.66843124384951480408832773143, −6.28382666553917981170316535815, −5.52171666837116827739117514921, −4.64475999839476868617412729710, −3.76225351956494245566245814105, −2.74071002283353410936321475172, −1.83344821327538226651826415780, −1.31628656141869859854913556079,
1.31628656141869859854913556079, 1.83344821327538226651826415780, 2.74071002283353410936321475172, 3.76225351956494245566245814105, 4.64475999839476868617412729710, 5.52171666837116827739117514921, 6.28382666553917981170316535815, 6.66843124384951480408832773143, 7.69203486300594974136712553247, 8.593790312985305814131505869791
