| L(s) = 1 | − 3-s − 3·5-s + 3·9-s − 9·13-s + 3·15-s − 3·17-s − 8·19-s − 9·23-s + 5·25-s − 8·27-s + 15·29-s + 8·31-s + 9·39-s + 15·41-s + 21·43-s − 9·45-s + 3·47-s + 2·49-s + 3·51-s + 3·53-s + 8·57-s + 3·59-s − 7·61-s + 27·65-s + 5·67-s + 9·69-s − 9·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 9-s − 2.49·13-s + 0.774·15-s − 0.727·17-s − 1.83·19-s − 1.87·23-s + 25-s − 1.53·27-s + 2.78·29-s + 1.43·31-s + 1.44·39-s + 2.34·41-s + 3.20·43-s − 1.34·45-s + 0.437·47-s + 2/7·49-s + 0.420·51-s + 0.412·53-s + 1.05·57-s + 0.390·59-s − 0.896·61-s + 3.34·65-s + 0.610·67-s + 1.08·69-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5926079802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5926079802\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 15 T + 104 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 15 T + 116 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−11.88896512764010894464710906566, −11.82743528073789792473562348404, −10.94205033080115522225951082749, −10.51339559977457695127757751045, −10.29478465606693746728075353088, −9.705512185149381066207179026617, −9.183919983071828966531807693445, −8.583611603569569972725368854051, −7.86595983306706129185814700038, −7.67740650569099686524916719349, −7.28316037914714979470581430316, −6.45318215565411345205655542251, −6.30606704157736832675000160426, −5.40974415982382139312497789140, −4.45219674863794293376985957698, −4.29054313203968733566774424446, −4.22110366608610118456235499223, −2.58149791104044373762974067924, −2.38393126347289137918503826864, −0.57258779718100859814409647740,
0.57258779718100859814409647740, 2.38393126347289137918503826864, 2.58149791104044373762974067924, 4.22110366608610118456235499223, 4.29054313203968733566774424446, 4.45219674863794293376985957698, 5.40974415982382139312497789140, 6.30606704157736832675000160426, 6.45318215565411345205655542251, 7.28316037914714979470581430316, 7.67740650569099686524916719349, 7.86595983306706129185814700038, 8.583611603569569972725368854051, 9.183919983071828966531807693445, 9.705512185149381066207179026617, 10.29478465606693746728075353088, 10.51339559977457695127757751045, 10.94205033080115522225951082749, 11.82743528073789792473562348404, 11.88896512764010894464710906566
