| L(s) = 1 | − 8·7-s − 9-s − 12·17-s − 8·23-s − 25-s + 16·31-s + 12·41-s + 24·47-s + 34·49-s + 8·63-s + 16·71-s − 20·73-s − 32·79-s + 81-s + 12·89-s + 36·97-s + 8·103-s − 12·113-s + 96·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 12·153-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 1/3·9-s − 2.91·17-s − 1.66·23-s − 1/5·25-s + 2.87·31-s + 1.87·41-s + 3.50·47-s + 34/7·49-s + 1.00·63-s + 1.89·71-s − 2.34·73-s − 3.60·79-s + 1/9·81-s + 1.27·89-s + 3.65·97-s + 0.788·103-s − 1.12·113-s + 8.80·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.970·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.039986841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039986841\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.834634286605556999566604427298, −8.489628515255200708701535321260, −7.910212424908324732022077052032, −7.46266654187780605512825787876, −7.07392267335978316812261189800, −6.73177270498444120422775643790, −6.45792657930084884431270107337, −6.05303798997080828314928789004, −5.90890625924389795865879289217, −5.68854929424284840006104864241, −4.53825439829316675696753470717, −4.46738477486952959806665076594, −4.14276338301519174485606026963, −3.66777009485718703081668030919, −3.07278845616675572720156459417, −2.74898010116590658983551316572, −2.38867596851675803254968842152, −2.04528636557942907441802336129, −0.67206654358324430554435033267, −0.45912073369796999403867732113,
0.45912073369796999403867732113, 0.67206654358324430554435033267, 2.04528636557942907441802336129, 2.38867596851675803254968842152, 2.74898010116590658983551316572, 3.07278845616675572720156459417, 3.66777009485718703081668030919, 4.14276338301519174485606026963, 4.46738477486952959806665076594, 4.53825439829316675696753470717, 5.68854929424284840006104864241, 5.90890625924389795865879289217, 6.05303798997080828314928789004, 6.45792657930084884431270107337, 6.73177270498444120422775643790, 7.07392267335978316812261189800, 7.46266654187780605512825787876, 7.910212424908324732022077052032, 8.489628515255200708701535321260, 8.834634286605556999566604427298
