L(s) = 1 | + (0.733 − 0.680i)3-s + (−0.826 + 0.563i)7-s + (0.0747 − 0.997i)9-s + (1.32 + 0.636i)13-s + (0.826 − 1.43i)19-s + (−0.222 + 0.974i)21-s + (0.826 + 0.563i)25-s + (−0.623 − 0.781i)27-s + (0.365 + 0.632i)31-s + (−0.722 − 1.84i)37-s + (1.40 − 0.432i)39-s + (0.0332 − 0.145i)43-s + (0.365 − 0.930i)49-s + (−0.367 − 1.61i)57-s + (0.455 + 1.16i)61-s + ⋯ |
L(s) = 1 | + (0.733 − 0.680i)3-s + (−0.826 + 0.563i)7-s + (0.0747 − 0.997i)9-s + (1.32 + 0.636i)13-s + (0.826 − 1.43i)19-s + (−0.222 + 0.974i)21-s + (0.826 + 0.563i)25-s + (−0.623 − 0.781i)27-s + (0.365 + 0.632i)31-s + (−0.722 − 1.84i)37-s + (1.40 − 0.432i)39-s + (0.0332 − 0.145i)43-s + (0.365 − 0.930i)49-s + (−0.367 − 1.61i)57-s + (0.455 + 1.16i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.506914142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.506914142\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.733 + 0.680i)T \) |
| 7 | \( 1 + (0.826 - 0.563i)T \) |
good | 5 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 11 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 13 | \( 1 + (-1.32 - 0.636i)T + (0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 19 | \( 1 + (-0.826 + 1.43i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 29 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.365 - 0.632i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.722 + 1.84i)T + (-0.733 + 0.680i)T^{2} \) |
| 41 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 43 | \( 1 + (-0.0332 + 0.145i)T + (-0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 53 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 59 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 61 | \( 1 + (-0.455 - 1.16i)T + (-0.733 + 0.680i)T^{2} \) |
| 67 | \( 1 + (-0.955 - 1.65i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (1.63 + 1.11i)T + (0.365 + 0.930i)T^{2} \) |
| 79 | \( 1 + (-0.365 + 0.632i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 89 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 97 | \( 1 + 1.80T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.874090878580027638049780505731, −8.664067642345714415979986127342, −7.37574602939598581886671921068, −6.89054499692784707197863449773, −6.15928069528780608330957561465, −5.28232419341985789127043257832, −3.97064458062080401176655201482, −3.20053830887219129048793847821, −2.40479910484953922330871911703, −1.14815439704503701051685649368,
1.36947168206067206064948540954, 2.90812956338175177005206522851, 3.48713809730931490642905624183, 4.19912961321106911424687257759, 5.27293486729236882620367034181, 6.14092081794669501930450030790, 6.95974331681810150981814835754, 8.098115973359098305923470634597, 8.292179513360728465731197039677, 9.386460887312235966953283828723