| L(s) = 1 | − 5-s − 11-s − 4·13-s − 6·17-s − 2·19-s − 9·23-s + 25-s − 9·29-s + 4·31-s + 2·37-s − 3·41-s − 11·43-s − 12·47-s + 55-s − 7·61-s + 4·65-s − 5·67-s − 6·71-s + 2·73-s + 16·79-s + 9·83-s + 6·85-s − 9·89-s + 2·95-s − 4·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.10·13-s − 1.45·17-s − 0.458·19-s − 1.87·23-s + 1/5·25-s − 1.67·29-s + 0.718·31-s + 0.328·37-s − 0.468·41-s − 1.67·43-s − 1.75·47-s + 0.134·55-s − 0.896·61-s + 0.496·65-s − 0.610·67-s − 0.712·71-s + 0.234·73-s + 1.80·79-s + 0.987·83-s + 0.650·85-s − 0.953·89-s + 0.205·95-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.79943866266079, −12.06465804623854, −11.84462540667768, −11.40975567672175, −10.89421017352108, −10.47219376208791, −9.869483965508554, −9.658970068093406, −9.099016514375847, −8.352919741281555, −8.293601888246557, −7.650088427536115, −7.227754038533836, −6.753503688525031, −6.160799350155437, −5.900926916428299, −5.013307733551060, −4.703142772190904, −4.365944748578777, −3.581038876098312, −3.312778537158132, −2.417854399898395, −2.051231859521791, −1.633848507412971, −0.4112895944739891, 0,
0.4112895944739891, 1.633848507412971, 2.051231859521791, 2.417854399898395, 3.312778537158132, 3.581038876098312, 4.365944748578777, 4.703142772190904, 5.013307733551060, 5.900926916428299, 6.160799350155437, 6.753503688525031, 7.227754038533836, 7.650088427536115, 8.293601888246557, 8.352919741281555, 9.099016514375847, 9.658970068093406, 9.869483965508554, 10.47219376208791, 10.89421017352108, 11.40975567672175, 11.84462540667768, 12.06465804623854, 12.79943866266079
