| L(s) = 1 | − 0.618·3-s + 1.90·5-s + 1.17·7-s − 0.618·9-s − 1.61·11-s − 1.17·15-s − 0.726·21-s + 1.90·23-s + 2.61·25-s + 27-s − 1.17·29-s + 1.00·33-s + 2.23·35-s + 1.17·37-s + 0.618·43-s − 1.17·45-s + 0.381·49-s − 3.07·55-s − 0.726·63-s − 1.17·69-s + 0.618·73-s − 1.61·75-s − 1.90·77-s + 0.726·87-s − 1.61·89-s − 97-s + 0.999·99-s + ⋯ |
| L(s) = 1 | − 0.618·3-s + 1.90·5-s + 1.17·7-s − 0.618·9-s − 1.61·11-s − 1.17·15-s − 0.726·21-s + 1.90·23-s + 2.61·25-s + 27-s − 1.17·29-s + 1.00·33-s + 2.23·35-s + 1.17·37-s + 0.618·43-s − 1.17·45-s + 0.381·49-s − 3.07·55-s − 0.726·63-s − 1.17·69-s + 0.618·73-s − 1.61·75-s − 1.90·77-s + 0.726·87-s − 1.61·89-s − 97-s + 0.999·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.471351063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.471351063\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 + 0.618T + T^{2} \) |
| 5 | \( 1 - 1.90T + T^{2} \) |
| 7 | \( 1 - 1.17T + T^{2} \) |
| 11 | \( 1 + 1.61T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - 1.90T + T^{2} \) |
| 29 | \( 1 + 1.17T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.17T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.618T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 0.618T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
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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
−8.954703446201520601902076244677, −8.188248028110388130480359521949, −7.34182668652713979329700221657, −6.41844317506333171614337114532, −5.50686430980829053919468564173, −5.36757934358516239986677384224, −4.67998348222682603225291530528, −2.88703602176581564913693923500, −2.31311234255746116173229814516, −1.21987291166824467592526166629,
1.21987291166824467592526166629, 2.31311234255746116173229814516, 2.88703602176581564913693923500, 4.67998348222682603225291530528, 5.36757934358516239986677384224, 5.50686430980829053919468564173, 6.41844317506333171614337114532, 7.34182668652713979329700221657, 8.188248028110388130480359521949, 8.954703446201520601902076244677
