L(s) = 1 | + 3-s − 4-s + 9-s − 12-s + 3·13-s − 3·16-s + 10·17-s − 6·19-s + 10·25-s + 27-s + 14·29-s + 10·31-s − 36-s + 3·39-s − 8·41-s + 4·47-s − 3·48-s − 6·49-s + 10·51-s − 3·52-s + 14·53-s − 6·57-s + 7·64-s + 10·67-s − 10·68-s + 4·73-s + 10·75-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s + 0.832·13-s − 3/4·16-s + 2.42·17-s − 1.37·19-s + 2·25-s + 0.192·27-s + 2.59·29-s + 1.79·31-s − 1/6·36-s + 0.480·39-s − 1.24·41-s + 0.583·47-s − 0.433·48-s − 6/7·49-s + 1.40·51-s − 0.416·52-s + 1.92·53-s − 0.794·57-s + 7/8·64-s + 1.22·67-s − 1.21·68-s + 0.468·73-s + 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2190591 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2190591 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.342073611\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.342073611\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( 1 - T \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( 1 - 10 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.075221418789761368343346778487, −7.21499588596995405117213146855, −6.81766197182721761545720981127, −6.45402147766387447850995117191, −6.25376400734509335170962073931, −5.38247812108768345438931433991, −5.08980211743778189427649356050, −4.68265910035935115520678378186, −4.14400513509503266006464924353, −3.75372157479712860091981475237, −3.05003637691719799877426895584, −2.83228603436609002409613108489, −2.17273163945289718543321501503, −1.10617251886235870643440312761, −0.910651937383598096124360068734,
0.910651937383598096124360068734, 1.10617251886235870643440312761, 2.17273163945289718543321501503, 2.83228603436609002409613108489, 3.05003637691719799877426895584, 3.75372157479712860091981475237, 4.14400513509503266006464924353, 4.68265910035935115520678378186, 5.08980211743778189427649356050, 5.38247812108768345438931433991, 6.25376400734509335170962073931, 6.45402147766387447850995117191, 6.81766197182721761545720981127, 7.21499588596995405117213146855, 8.075221418789761368343346778487