| L(s) = 1 | + 4-s − 5-s − 7-s − 11-s + 13-s + 16-s + 17-s − 20-s + 25-s − 28-s + 2·29-s + 35-s − 44-s + 47-s + 49-s + 52-s + 55-s + 64-s − 65-s + 68-s − 71-s + 73-s + 77-s − 79-s − 80-s + 83-s − 85-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s − 7-s − 11-s + 13-s + 16-s + 17-s − 20-s + 25-s − 28-s + 2·29-s + 35-s − 44-s + 47-s + 49-s + 52-s + 55-s + 64-s − 65-s + 68-s − 71-s + 73-s + 77-s − 79-s − 80-s + 83-s − 85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2835 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.211937684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.211937684\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.776050160956307745027415860733, −8.083788478046481921641957064421, −7.48679236827280726651203988365, −6.71748603204775596730389659595, −6.08036631942692139265072587790, −5.21720738110119748122950866701, −4.01425962109183598008855810593, −3.20093359630550995920967567812, −2.65169533806144126059627921593, −1.02549952861216185809098240705,
1.02549952861216185809098240705, 2.65169533806144126059627921593, 3.20093359630550995920967567812, 4.01425962109183598008855810593, 5.21720738110119748122950866701, 6.08036631942692139265072587790, 6.71748603204775596730389659595, 7.48679236827280726651203988365, 8.083788478046481921641957064421, 8.776050160956307745027415860733
