| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s − 2·7-s + 8-s − 10-s + 4·11-s + 3·13-s + 2·14-s − 15-s − 16-s − 4·17-s + 8·19-s + 2·21-s − 4·22-s + 6·23-s − 24-s − 3·26-s + 27-s + 10·29-s + 30-s + 2·31-s − 4·33-s + 4·34-s − 2·35-s − 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.832·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 1.83·19-s + 0.436·21-s − 0.852·22-s + 1.25·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s + 1.85·29-s + 0.182·30-s + 0.359·31-s − 0.696·33-s + 0.685·34-s − 0.338·35-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.119871739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119871739\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2 T - 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.82968777311078803666024264811, −10.49141368337666326423582416115, −9.916436371647121268699902052222, −9.795021897069123410852039953444, −9.078593819957897214792129023348, −8.840088507065860518693972175046, −8.655238444588747556259208083972, −7.895621802286099964824644110513, −7.22744542919858744365105500972, −6.88981659299806373152911804805, −6.47373468576235738619693083457, −6.10634893588634323077240572864, −5.52483200491050289155150484852, −4.83999106076969052721781158083, −4.57929764095546905524834171997, −3.43438431865943422985268001112, −3.42636963520616200921620401874, −2.40450431160513422748603935096, −1.37480648026459237086670625954, −0.809873558801618410664511583089,
0.809873558801618410664511583089, 1.37480648026459237086670625954, 2.40450431160513422748603935096, 3.42636963520616200921620401874, 3.43438431865943422985268001112, 4.57929764095546905524834171997, 4.83999106076969052721781158083, 5.52483200491050289155150484852, 6.10634893588634323077240572864, 6.47373468576235738619693083457, 6.88981659299806373152911804805, 7.22744542919858744365105500972, 7.895621802286099964824644110513, 8.655238444588747556259208083972, 8.840088507065860518693972175046, 9.078593819957897214792129023348, 9.795021897069123410852039953444, 9.916436371647121268699902052222, 10.49141368337666326423582416115, 10.82968777311078803666024264811
