L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 6·9-s − 8·11-s + 5·16-s − 12·18-s − 16·22-s − 8·29-s + 6·32-s − 18·36-s − 12·37-s − 24·43-s − 24·44-s − 24·53-s − 16·58-s + 7·64-s − 24·67-s + 16·71-s − 24·72-s − 24·74-s + 27·81-s − 48·86-s − 32·88-s + 48·99-s − 48·106-s + 24·107-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2·9-s − 2.41·11-s + 5/4·16-s − 2.82·18-s − 3.41·22-s − 1.48·29-s + 1.06·32-s − 3·36-s − 1.97·37-s − 3.65·43-s − 3.61·44-s − 3.29·53-s − 2.10·58-s + 7/8·64-s − 2.93·67-s + 1.89·71-s − 2.82·72-s − 2.78·74-s + 3·81-s − 5.17·86-s − 3.41·88-s + 4.82·99-s − 4.66·106-s + 2.32·107-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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−8.404936149144031677513775353019, −8.363890546288383534446644373507, −7.951527539037869539182624015648, −7.64676227678662275011660042611, −7.13864555412994918477703332759, −6.70412314992962757671132540921, −6.21616518601793966339405085165, −5.93237179282087313858039130507, −5.37306626198236309732120880153, −5.28414901579079646322877809444, −4.89804885205522541521761378318, −4.63838188154805417797408088974, −3.64122797614887584500242126553, −3.39007701798454149535875122819, −2.97785655923287429873323717644, −2.79079397579378826809843370943, −1.88171923399790662800702456563, −1.84006062092222343779552731877, 0, 0,
1.84006062092222343779552731877, 1.88171923399790662800702456563, 2.79079397579378826809843370943, 2.97785655923287429873323717644, 3.39007701798454149535875122819, 3.64122797614887584500242126553, 4.63838188154805417797408088974, 4.89804885205522541521761378318, 5.28414901579079646322877809444, 5.37306626198236309732120880153, 5.93237179282087313858039130507, 6.21616518601793966339405085165, 6.70412314992962757671132540921, 7.13864555412994918477703332759, 7.64676227678662275011660042611, 7.951527539037869539182624015648, 8.363890546288383534446644373507, 8.404936149144031677513775353019