| L(s) = 1 | − 3-s − 4-s − 5·7-s + 9-s + 12-s + 13-s + 16-s + 19-s + 5·21-s + 25-s − 27-s + 5·28-s + 13·31-s − 36-s + 4·37-s − 39-s − 17·43-s − 48-s + 7·49-s − 52-s − 57-s − 10·61-s − 5·63-s − 64-s + 5·67-s − 12·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 1.88·7-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.229·19-s + 1.09·21-s + 1/5·25-s − 0.192·27-s + 0.944·28-s + 2.33·31-s − 1/6·36-s + 0.657·37-s − 0.160·39-s − 2.59·43-s − 0.144·48-s + 49-s − 0.138·52-s − 0.132·57-s − 1.28·61-s − 0.629·63-s − 1/8·64-s + 0.610·67-s − 1.40·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6957031669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6957031669\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 95 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.634769676691059675473283175102, −8.117894573455935801809461203028, −7.70395100151010339738107783667, −7.02050367740557281051558463121, −6.55684705828307096003144671074, −6.27593584172535976829232752166, −6.03073677027748666534747854795, −5.14539291742820593469727213880, −4.89265680110243887977130886450, −4.21792046913955212476581047153, −3.62438617236192791170932718197, −3.12185107973177026266918502731, −2.65851797802163047719404235723, −1.46899302767754977256381095776, −0.48431574800842560447758169055,
0.48431574800842560447758169055, 1.46899302767754977256381095776, 2.65851797802163047719404235723, 3.12185107973177026266918502731, 3.62438617236192791170932718197, 4.21792046913955212476581047153, 4.89265680110243887977130886450, 5.14539291742820593469727213880, 6.03073677027748666534747854795, 6.27593584172535976829232752166, 6.55684705828307096003144671074, 7.02050367740557281051558463121, 7.70395100151010339738107783667, 8.117894573455935801809461203028, 8.634769676691059675473283175102
