| L(s) = 1 | + 1.53·3-s + 4-s − 5-s − 1.87·7-s + 1.34·9-s + 1.53·12-s + 0.347·13-s − 1.53·15-s + 16-s − 20-s − 2.87·21-s − 1.87·23-s + 0.532·27-s − 1.87·28-s + 0.347·29-s + 1.87·35-s + 1.34·36-s + 1.53·37-s + 0.532·39-s − 1.87·41-s − 43-s − 1.34·45-s + 0.347·47-s + 1.53·48-s + 2.53·49-s + 0.347·52-s + 1.53·59-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 4-s − 5-s − 1.87·7-s + 1.34·9-s + 1.53·12-s + 0.347·13-s − 1.53·15-s + 16-s − 20-s − 2.87·21-s − 1.87·23-s + 0.532·27-s − 1.87·28-s + 0.347·29-s + 1.87·35-s + 1.34·36-s + 1.53·37-s + 0.532·39-s − 1.87·41-s − 43-s − 1.34·45-s + 0.347·47-s + 1.53·48-s + 2.53·49-s + 0.347·52-s + 1.53·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.189133067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.189133067\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 419 | \( 1 - T \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - 1.53T + T^{2} \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + 1.87T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.347T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.87T + T^{2} \) |
| 29 | \( 1 - 0.347T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.53T + T^{2} \) |
| 41 | \( 1 + 1.87T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - 0.347T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.53T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 - 1.53T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−11.58218664277438932648958977460, −10.19803307288361973313518359017, −9.704297221093124693408962466085, −8.504327039534987505992907519082, −7.82779232808556939921853367999, −6.92321772573254007220961298862, −6.09566299615662881995283829554, −3.85053599856685108779614652121, −3.36735447238549804545302550662, −2.32907613575405534117182606719,
2.32907613575405534117182606719, 3.36735447238549804545302550662, 3.85053599856685108779614652121, 6.09566299615662881995283829554, 6.92321772573254007220961298862, 7.82779232808556939921853367999, 8.504327039534987505992907519082, 9.704297221093124693408962466085, 10.19803307288361973313518359017, 11.58218664277438932648958977460
