L(s) = 1 | + 2·3-s + 2·5-s + 2·7-s + 3·9-s + 4·13-s + 4·15-s + 4·17-s + 4·21-s + 8·23-s + 3·25-s + 4·27-s + 4·29-s + 4·35-s + 12·37-s + 8·39-s − 4·41-s − 8·43-s + 6·45-s + 3·49-s + 8·51-s − 4·53-s + 4·61-s + 6·63-s + 8·65-s − 8·67-s + 16·69-s + 16·71-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 1.10·13-s + 1.03·15-s + 0.970·17-s + 0.872·21-s + 1.66·23-s + 3/5·25-s + 0.769·27-s + 0.742·29-s + 0.676·35-s + 1.97·37-s + 1.28·39-s − 0.624·41-s − 1.21·43-s + 0.894·45-s + 3/7·49-s + 1.12·51-s − 0.549·53-s + 0.512·61-s + 0.755·63-s + 0.992·65-s − 0.977·67-s + 1.92·69-s + 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(10.76637120\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.76637120\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 286 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−8.120151440435996492672146225949, −8.015587522875009855034829938933, −7.50572990328387576277108176387, −7.17587323374102250790035993067, −6.72130538508751416413037371588, −6.55363185795382788588778725664, −5.90751808942305759661568065561, −5.85829997836559913129376573172, −5.20442490003475944838325789943, −4.95767193189019224858110557523, −4.60786698437838234935026578794, −4.16754368742790217948686106290, −3.68319029288052364716594794762, −3.28556491636812261778538741393, −2.86546997515383267152171137862, −2.71088875382729256610936253916, −1.87110344853875908263705189823, −1.77458238495113806394634209507, −1.02373242603094179679415200563, −0.904682239764224636291642222556,
0.904682239764224636291642222556, 1.02373242603094179679415200563, 1.77458238495113806394634209507, 1.87110344853875908263705189823, 2.71088875382729256610936253916, 2.86546997515383267152171137862, 3.28556491636812261778538741393, 3.68319029288052364716594794762, 4.16754368742790217948686106290, 4.60786698437838234935026578794, 4.95767193189019224858110557523, 5.20442490003475944838325789943, 5.85829997836559913129376573172, 5.90751808942305759661568065561, 6.55363185795382788588778725664, 6.72130538508751416413037371588, 7.17587323374102250790035993067, 7.50572990328387576277108176387, 8.015587522875009855034829938933, 8.120151440435996492672146225949