| L(s) = 1 | − 3-s + 9-s + 3·11-s − 3·17-s − 16·19-s + 25-s − 27-s − 3·33-s − 3·41-s − 4·43-s − 10·49-s + 3·51-s + 16·57-s + 6·59-s − 10·67-s − 5·73-s − 75-s + 81-s + 27·89-s + 97-s + 3·99-s + 3·107-s + 3·113-s + 5·121-s + 3·123-s + 127-s + 4·129-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.904·11-s − 0.727·17-s − 3.67·19-s + 1/5·25-s − 0.192·27-s − 0.522·33-s − 0.468·41-s − 0.609·43-s − 1.42·49-s + 0.420·51-s + 2.11·57-s + 0.781·59-s − 1.22·67-s − 0.585·73-s − 0.115·75-s + 1/9·81-s + 2.86·89-s + 0.101·97-s + 0.301·99-s + 0.290·107-s + 0.282·113-s + 5/11·121-s + 0.270·123-s + 0.0887·127-s + 0.352·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7764291767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7764291767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78899564259219579177095851648, −7.12004499811995814832830992412, −6.73982400433838448566383704116, −6.44688400781234932341436099544, −6.19208644340779983296691058941, −5.78549076916501490019141600451, −4.99095465856288090592337941258, −4.61418337716936210466403747668, −4.35767752594163859964945927842, −3.82056981517042218834564107265, −3.35687557768969660990438456180, −2.45841532863111583796640045388, −2.01934176892312000779111410566, −1.50753100204363295324553896843, −0.36103444027355130171359211061,
0.36103444027355130171359211061, 1.50753100204363295324553896843, 2.01934176892312000779111410566, 2.45841532863111583796640045388, 3.35687557768969660990438456180, 3.82056981517042218834564107265, 4.35767752594163859964945927842, 4.61418337716936210466403747668, 4.99095465856288090592337941258, 5.78549076916501490019141600451, 6.19208644340779983296691058941, 6.44688400781234932341436099544, 6.73982400433838448566383704116, 7.12004499811995814832830992412, 7.78899564259219579177095851648
