L(s) = 1 | − 2·5-s + 2·11-s + 2·13-s − 6·17-s + 19-s + 2·23-s − 25-s − 4·29-s + 8·31-s − 2·37-s + 8·41-s + 8·43-s + 2·47-s − 7·49-s + 4·53-s − 4·55-s + 2·61-s − 4·65-s − 12·67-s − 4·71-s + 6·73-s + 16·79-s + 6·83-s + 12·85-s − 2·95-s − 2·97-s + 10·101-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.603·11-s + 0.554·13-s − 1.45·17-s + 0.229·19-s + 0.417·23-s − 1/5·25-s − 0.742·29-s + 1.43·31-s − 0.328·37-s + 1.24·41-s + 1.21·43-s + 0.291·47-s − 49-s + 0.549·53-s − 0.539·55-s + 0.256·61-s − 0.496·65-s − 1.46·67-s − 0.474·71-s + 0.702·73-s + 1.80·79-s + 0.658·83-s + 1.30·85-s − 0.205·95-s − 0.203·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.429743256\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.429743256\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.873111295613709275849760694334, −8.044021726622219486067451800232, −7.36880369981808824510206698261, −6.56305934461778732377407264371, −5.88098732566068880719234635868, −4.66462432003445626826076343394, −4.11823581145492617961933321768, −3.26694707001641266093809028834, −2.11490321724667271499625311110, −0.74382195135806404149682007413,
0.74382195135806404149682007413, 2.11490321724667271499625311110, 3.26694707001641266093809028834, 4.11823581145492617961933321768, 4.66462432003445626826076343394, 5.88098732566068880719234635868, 6.56305934461778732377407264371, 7.36880369981808824510206698261, 8.044021726622219486067451800232, 8.873111295613709275849760694334