| L(s) = 1 | − 4-s − 4·5-s − 9-s + 16-s + 4·19-s + 4·20-s + 11·25-s + 16·31-s + 36-s + 12·41-s + 4·45-s + 10·49-s − 12·59-s + 16·61-s − 64-s + 16·71-s − 4·76-s + 32·79-s − 4·80-s + 81-s − 12·89-s − 16·95-s − 11·100-s − 20·101-s + 24·109-s − 22·121-s − 16·124-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.78·5-s − 1/3·9-s + 1/4·16-s + 0.917·19-s + 0.894·20-s + 11/5·25-s + 2.87·31-s + 1/6·36-s + 1.87·41-s + 0.596·45-s + 10/7·49-s − 1.56·59-s + 2.04·61-s − 1/8·64-s + 1.89·71-s − 0.458·76-s + 3.60·79-s − 0.447·80-s + 1/9·81-s − 1.27·89-s − 1.64·95-s − 1.09·100-s − 1.99·101-s + 2.29·109-s − 2·121-s − 1.43·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.283239077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283239077\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.949794568999077430049025979163, −9.687900147503881946912963157314, −9.002733054923657741675495751317, −8.929871208405485281868573329514, −8.168879983998667802800327860234, −8.061045071109727557833480120655, −7.74442124425856362625932106275, −7.29752072306438924333091167403, −6.68001892301503541499825680396, −6.41770390933790760297040453226, −5.79279481821316332130162607663, −5.03142468336652513445714619554, −4.99065158219781470986666377204, −4.24858544144779233391763768170, −3.93008288872750426353207224817, −3.52841878228554168905291947366, −2.77400149945224185587986444726, −2.50563471969673177083762867168, −1.04900108513294705733795950318, −0.64820559882020024916121422129,
0.64820559882020024916121422129, 1.04900108513294705733795950318, 2.50563471969673177083762867168, 2.77400149945224185587986444726, 3.52841878228554168905291947366, 3.93008288872750426353207224817, 4.24858544144779233391763768170, 4.99065158219781470986666377204, 5.03142468336652513445714619554, 5.79279481821316332130162607663, 6.41770390933790760297040453226, 6.68001892301503541499825680396, 7.29752072306438924333091167403, 7.74442124425856362625932106275, 8.061045071109727557833480120655, 8.168879983998667802800327860234, 8.929871208405485281868573329514, 9.002733054923657741675495751317, 9.687900147503881946912963157314, 9.949794568999077430049025979163
