| L(s) = 1 | + (−0.154 + 1.40i)2-s + (−1.95 − 0.433i)4-s − 0.162·5-s + i·7-s + (0.909 − 2.67i)8-s + (0.0249 − 0.227i)10-s + 2.40i·11-s − 4.76i·13-s + (−1.40 − 0.154i)14-s + (3.62 + 1.69i)16-s − 2.74i·17-s − 1.86·19-s + (0.316 + 0.0701i)20-s + (−3.37 − 0.370i)22-s + 1.32·23-s + ⋯ |
| L(s) = 1 | + (−0.108 + 0.994i)2-s + (−0.976 − 0.216i)4-s − 0.0724·5-s + 0.377i·7-s + (0.321 − 0.946i)8-s + (0.00789 − 0.0720i)10-s + 0.724i·11-s − 1.32i·13-s + (−0.375 − 0.0411i)14-s + (0.906 + 0.422i)16-s − 0.665i·17-s − 0.426·19-s + (0.0707 + 0.0156i)20-s + (−0.720 − 0.0789i)22-s + 0.277·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9698905433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9698905433\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.154 - 1.40i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 0.162T + 5T^{2} \) |
| 11 | \( 1 - 2.40iT - 11T^{2} \) |
| 13 | \( 1 + 4.76iT - 13T^{2} \) |
| 17 | \( 1 + 2.74iT - 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 - 1.32T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 31 | \( 1 + 8.01iT - 31T^{2} \) |
| 37 | \( 1 + 6.09iT - 37T^{2} \) |
| 41 | \( 1 + 3.16iT - 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 8.22T + 53T^{2} \) |
| 59 | \( 1 - 5.45iT - 59T^{2} \) |
| 61 | \( 1 + 4.28iT - 61T^{2} \) |
| 67 | \( 1 + 4.61T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 + 9.75iT - 79T^{2} \) |
| 83 | \( 1 + 16.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.35iT - 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−9.293822641541331468339027093470, −8.568422672568547988551495712376, −7.56286502753403340835310379634, −7.32266269081369188510077165856, −5.99870905764427839102102792604, −5.56271606833921736107185089462, −4.56836635633710974685347052962, −3.64108226663632648322399887620, −2.25071345204246428244337411366, −0.44069829389018379992458698936,
1.23412195973925872732716038748, 2.30722104688566803374371877946, 3.56188018615769083787870719751, 4.15043733433856366845308503271, 5.15892092050796700990524792478, 6.21023947830075988486478183881, 7.19042043289757405513319396556, 8.228061006769069263007583586883, 8.820340179804446244254076787935, 9.578886524549320553410914531759
