| L(s) = 1 | + 3·3-s + 5-s + 4·7-s + 6·9-s − 11-s + 3·15-s − 2·17-s + 14·19-s + 12·21-s − 6·23-s + 9·27-s + 2·29-s + 10·31-s − 3·33-s + 4·35-s + 16·37-s − 5·41-s − 5·43-s + 6·45-s + 4·47-s + 7·49-s − 6·51-s − 20·53-s − 55-s + 42·57-s + 9·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 1.51·7-s + 2·9-s − 0.301·11-s + 0.774·15-s − 0.485·17-s + 3.21·19-s + 2.61·21-s − 1.25·23-s + 1.73·27-s + 0.371·29-s + 1.79·31-s − 0.522·33-s + 0.676·35-s + 2.63·37-s − 0.780·41-s − 0.762·43-s + 0.894·45-s + 0.583·47-s + 49-s − 0.840·51-s − 2.74·53-s − 0.134·55-s + 5.56·57-s + 1.17·59-s + 1.28·61-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
| \(L(1)\) |
\(\approx\) |
\(7.240154738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.240154738\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4 T - 67 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.798351062923126448121131704690, −9.535848300184255122605227320524, −8.712148427351964475771060359653, −8.508959988498463799152126065969, −8.087847702906179405575879192321, −7.917147713502018798668098894394, −7.37913611746541974496243430497, −7.30198898408694589929224545003, −6.51045819457831156908692603978, −6.06955489432897134233643096259, −5.46459458287792361184142916905, −5.08613778708341914565026245301, −4.45729376076612660939166826063, −4.39974970225123159878759651307, −3.53825745495048976239772471053, −3.13773298847982114725143311162, −2.55526421771305718533387832471, −2.28504155336601607978579438023, −1.28364590297471768186510535734, −1.27146389956915775190390834897,
1.27146389956915775190390834897, 1.28364590297471768186510535734, 2.28504155336601607978579438023, 2.55526421771305718533387832471, 3.13773298847982114725143311162, 3.53825745495048976239772471053, 4.39974970225123159878759651307, 4.45729376076612660939166826063, 5.08613778708341914565026245301, 5.46459458287792361184142916905, 6.06955489432897134233643096259, 6.51045819457831156908692603978, 7.30198898408694589929224545003, 7.37913611746541974496243430497, 7.917147713502018798668098894394, 8.087847702906179405575879192321, 8.508959988498463799152126065969, 8.712148427351964475771060359653, 9.535848300184255122605227320524, 9.798351062923126448121131704690
