| L(s) = 1 | − 4-s + 9-s − 4·11-s + 16-s + 12·19-s − 2·29-s + 2·31-s − 36-s − 6·41-s + 4·44-s + 49-s + 8·59-s − 4·61-s − 64-s + 2·71-s − 12·76-s + 16·79-s − 8·81-s − 2·89-s − 4·99-s + 22·101-s + 12·109-s + 2·116-s − 121-s − 2·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/3·9-s − 1.20·11-s + 1/4·16-s + 2.75·19-s − 0.371·29-s + 0.359·31-s − 1/6·36-s − 0.937·41-s + 0.603·44-s + 1/7·49-s + 1.04·59-s − 0.512·61-s − 1/8·64-s + 0.237·71-s − 1.37·76-s + 1.80·79-s − 8/9·81-s − 0.211·89-s − 0.402·99-s + 2.18·101-s + 1.14·109-s + 0.185·116-s − 0.0909·121-s − 0.179·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.332550857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.332550857\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.437382730695331110809339994766, −9.029725241534180990007378974216, −8.443110855743129358253376452889, −7.75001128763125133211871317500, −7.69070526066147698637407976828, −7.06843382677037949556666924223, −6.47272947525813797860142470791, −5.61215466967589650670454879370, −5.34867949786088770626255565334, −4.90287105263978612276724574383, −4.19978076136144134659613644444, −3.34927270683615800454886406427, −3.02688698319208599371399938902, −1.98086255952639771007348905145, −0.847625250635756278445922102972,
0.847625250635756278445922102972, 1.98086255952639771007348905145, 3.02688698319208599371399938902, 3.34927270683615800454886406427, 4.19978076136144134659613644444, 4.90287105263978612276724574383, 5.34867949786088770626255565334, 5.61215466967589650670454879370, 6.47272947525813797860142470791, 7.06843382677037949556666924223, 7.69070526066147698637407976828, 7.75001128763125133211871317500, 8.443110855743129358253376452889, 9.029725241534180990007378974216, 9.437382730695331110809339994766
