L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 4·9-s − 4·11-s + 5·16-s + 8·18-s + 8·22-s + 8·23-s + 4·29-s − 6·32-s − 12·36-s − 20·37-s − 4·43-s − 12·44-s − 16·46-s + 4·53-s − 8·58-s + 7·64-s − 24·67-s − 24·71-s + 16·72-s + 40·74-s − 8·79-s + 7·81-s + 8·86-s + 16·88-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 4/3·9-s − 1.20·11-s + 5/4·16-s + 1.88·18-s + 1.70·22-s + 1.66·23-s + 0.742·29-s − 1.06·32-s − 2·36-s − 3.28·37-s − 0.609·43-s − 1.80·44-s − 2.35·46-s + 0.549·53-s − 1.05·58-s + 7/8·64-s − 2.93·67-s − 2.84·71-s + 1.88·72-s + 4.64·74-s − 0.900·79-s + 7/9·81-s + 0.862·86-s + 1.70·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 144 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
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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
−8.784467759916860109698174652067, −8.515948099688136973439282906171, −8.142749811253533413828290341214, −7.61880032922231220197564201244, −7.21993130346444098396225189624, −7.17201756060828013185414295338, −6.45997921700750033789536909292, −6.21407898344471964085582248513, −5.49870623595802656801151655402, −5.48231713596203108958680423804, −4.89724914111012723742836738196, −4.50446129371836406425191915992, −3.47519687889433557163866123930, −3.29999362433018355676864180741, −2.65325475390506882403929399354, −2.58572925275752240835527945302, −1.64863925254873572140399482410, −1.22729590705946039541500637626, 0, 0,
1.22729590705946039541500637626, 1.64863925254873572140399482410, 2.58572925275752240835527945302, 2.65325475390506882403929399354, 3.29999362433018355676864180741, 3.47519687889433557163866123930, 4.50446129371836406425191915992, 4.89724914111012723742836738196, 5.48231713596203108958680423804, 5.49870623595802656801151655402, 6.21407898344471964085582248513, 6.45997921700750033789536909292, 7.17201756060828013185414295338, 7.21993130346444098396225189624, 7.61880032922231220197564201244, 8.142749811253533413828290341214, 8.515948099688136973439282906171, 8.784467759916860109698174652067