| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s + 16-s − 3·17-s − 2·19-s + 3·22-s − 25-s − 32-s + 3·34-s + 2·38-s − 9·41-s + 7·43-s − 3·44-s − 10·49-s + 50-s − 15·59-s + 64-s + 10·67-s − 3·68-s − 14·73-s − 2·76-s + 9·82-s + 21·83-s − 7·86-s + 3·88-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s + 1/4·16-s − 0.727·17-s − 0.458·19-s + 0.639·22-s − 1/5·25-s − 0.176·32-s + 0.514·34-s + 0.324·38-s − 1.40·41-s + 1.06·43-s − 0.452·44-s − 1.42·49-s + 0.141·50-s − 1.95·59-s + 1/8·64-s + 1.22·67-s − 0.363·68-s − 1.63·73-s − 0.229·76-s + 0.993·82-s + 2.30·83-s − 0.754·86-s + 0.319·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.344472008502723218469071659827, −8.969236334803049982117562754743, −8.249120060014703418838716811920, −8.044931508328250774398122649158, −7.51393230366854168431436267607, −6.81850674748621907350392229252, −6.49249417251469394022068181847, −5.83937992260413585314544263533, −5.20419150880199905972121759941, −4.64813983258769297069089556777, −3.90495047201058901864390011350, −3.06799025366596494978306712499, −2.42280584847596942244995624263, −1.57593862833188896251482267618, 0,
1.57593862833188896251482267618, 2.42280584847596942244995624263, 3.06799025366596494978306712499, 3.90495047201058901864390011350, 4.64813983258769297069089556777, 5.20419150880199905972121759941, 5.83937992260413585314544263533, 6.49249417251469394022068181847, 6.81850674748621907350392229252, 7.51393230366854168431436267607, 8.044931508328250774398122649158, 8.249120060014703418838716811920, 8.969236334803049982117562754743, 9.344472008502723218469071659827
