| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s + 8·17-s + 12·19-s − 4·23-s + 14·31-s + 6·32-s + 16·34-s + 24·38-s − 8·46-s + 4·47-s + 5·49-s − 6·53-s − 8·61-s + 28·62-s + 7·64-s + 24·68-s + 36·76-s − 10·83-s − 12·92-s + 8·94-s + 10·98-s − 12·106-s + 30·107-s + 20·109-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 5/4·16-s + 1.94·17-s + 2.75·19-s − 0.834·23-s + 2.51·31-s + 1.06·32-s + 2.74·34-s + 3.89·38-s − 1.17·46-s + 0.583·47-s + 5/7·49-s − 0.824·53-s − 1.02·61-s + 3.55·62-s + 7/8·64-s + 2.91·68-s + 4.12·76-s − 1.09·83-s − 1.25·92-s + 0.825·94-s + 1.01·98-s − 1.16·106-s + 2.90·107-s + 1.91·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.583417374\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.583417374\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 127 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 175 T^{2} + p^{2} T^{4} \) |
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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
−9.831766250536615909555991793629, −9.745710883923792087820376688981, −8.999396774747180827227301432003, −8.521640902459070146262123743588, −7.80286480238678608669437815722, −7.76977121011768601913358247835, −7.41152490360258934930589333356, −6.92491019420454483770102890154, −6.23668807634405041476143501819, −6.03490131305999332310631451392, −5.60978920086988969726692862806, −5.12800961747859868254216974970, −4.85036360342555564090829170734, −4.31693309753682224821546583327, −3.53307271168479685023107989173, −3.48883838002187822386271183003, −2.80562676525849665288697615059, −2.48726936281705293448809254923, −1.31472464666321829271225485775, −1.09564415871161753804012423175,
1.09564415871161753804012423175, 1.31472464666321829271225485775, 2.48726936281705293448809254923, 2.80562676525849665288697615059, 3.48883838002187822386271183003, 3.53307271168479685023107989173, 4.31693309753682224821546583327, 4.85036360342555564090829170734, 5.12800961747859868254216974970, 5.60978920086988969726692862806, 6.03490131305999332310631451392, 6.23668807634405041476143501819, 6.92491019420454483770102890154, 7.41152490360258934930589333356, 7.76977121011768601913358247835, 7.80286480238678608669437815722, 8.521640902459070146262123743588, 8.999396774747180827227301432003, 9.745710883923792087820376688981, 9.831766250536615909555991793629
