L(s) = 1 | − 2·5-s + 8·11-s − 6·13-s + 12·17-s + 8·19-s + 2·25-s − 6·29-s + 8·31-s − 2·37-s + 8·43-s + 16·47-s − 2·49-s + 14·53-s − 16·55-s + 6·61-s + 12·65-s − 16·67-s − 24·79-s − 8·83-s − 24·85-s − 16·95-s + 16·97-s + 14·101-s − 16·107-s − 10·109-s + 32·113-s + 32·121-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 2.41·11-s − 1.66·13-s + 2.91·17-s + 1.83·19-s + 2/5·25-s − 1.11·29-s + 1.43·31-s − 0.328·37-s + 1.21·43-s + 2.33·47-s − 2/7·49-s + 1.92·53-s − 2.15·55-s + 0.768·61-s + 1.48·65-s − 1.95·67-s − 2.70·79-s − 0.878·83-s − 2.60·85-s − 1.64·95-s + 1.62·97-s + 1.39·101-s − 1.54·107-s − 0.957·109-s + 3.01·113-s + 2.90·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.700911238\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.700911238\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.664777297859929740659437856465, −7.83325522394818634085343838664, −7.67837427351456774870628049411, −7.51429224173958393752440525860, −7.05469919678144182569605052107, −7.02694324900049830732507678505, −6.29741948909370274433204702134, −5.82878137476376961810194530978, −5.47134438700881188236844577146, −5.44025664860789588903894374593, −4.68331474333806996471422034994, −4.16867341776349188836633158924, −4.13317443937412625674993055332, −3.52885920743695155344578371168, −3.17909892509341223686192717639, −2.87940617735897175679002636910, −2.21054636841203996196304650140, −1.40604506014445785688187876719, −1.05434687190868088777362237911, −0.67235618725128835732445613211,
0.67235618725128835732445613211, 1.05434687190868088777362237911, 1.40604506014445785688187876719, 2.21054636841203996196304650140, 2.87940617735897175679002636910, 3.17909892509341223686192717639, 3.52885920743695155344578371168, 4.13317443937412625674993055332, 4.16867341776349188836633158924, 4.68331474333806996471422034994, 5.44025664860789588903894374593, 5.47134438700881188236844577146, 5.82878137476376961810194530978, 6.29741948909370274433204702134, 7.02694324900049830732507678505, 7.05469919678144182569605052107, 7.51429224173958393752440525860, 7.67837427351456774870628049411, 7.83325522394818634085343838664, 8.664777297859929740659437856465