L(s) = 1 | + (−0.433 + 0.900i)3-s + (0.974 − 0.222i)4-s + (0.623 − 0.781i)7-s + (−0.623 − 0.781i)9-s + (−0.222 + 0.974i)12-s + (0.433 + 0.900i)13-s + (0.900 − 0.433i)16-s + (−1.40 + 1.40i)19-s + (0.433 + 0.900i)21-s + (0.781 − 0.623i)25-s + (0.974 − 0.222i)27-s + (0.433 − 0.900i)28-s + (1.19 − 1.19i)31-s + (−0.781 − 0.623i)36-s + (0.900 + 1.43i)37-s + ⋯ |
L(s) = 1 | + (−0.433 + 0.900i)3-s + (0.974 − 0.222i)4-s + (0.623 − 0.781i)7-s + (−0.623 − 0.781i)9-s + (−0.222 + 0.974i)12-s + (0.433 + 0.900i)13-s + (0.900 − 0.433i)16-s + (−1.40 + 1.40i)19-s + (0.433 + 0.900i)21-s + (0.781 − 0.623i)25-s + (0.974 − 0.222i)27-s + (0.433 − 0.900i)28-s + (1.19 − 1.19i)31-s + (−0.781 − 0.623i)36-s + (0.900 + 1.43i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.342627031\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.342627031\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.433 - 0.900i)T \) |
| 7 | \( 1 + (-0.623 + 0.781i)T \) |
| 13 | \( 1 + (-0.433 - 0.900i)T \) |
good | 2 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 5 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 11 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 17 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 19 | \( 1 + (1.40 - 1.40i)T - iT^{2} \) |
| 23 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 29 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 31 | \( 1 + (-1.19 + 1.19i)T - iT^{2} \) |
| 37 | \( 1 + (-0.900 - 1.43i)T + (-0.433 + 0.900i)T^{2} \) |
| 41 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 43 | \( 1 + (0.678 + 1.40i)T + (-0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 53 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 59 | \( 1 + (-0.781 - 0.623i)T^{2} \) |
| 61 | \( 1 + (0.433 + 0.0990i)T + (0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 + (-1.33 - 1.33i)T + iT^{2} \) |
| 71 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 73 | \( 1 + (0.119 - 1.05i)T + (-0.974 - 0.222i)T^{2} \) |
| 79 | \( 1 + 1.94T + T^{2} \) |
| 83 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 89 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 97 | \( 1 + (0.158 - 0.158i)T - iT^{2} \) |
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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
−9.823818765136839813808699820228, −8.545206153869274089426487517951, −8.046395541746476667566889293607, −6.78689222355352354865459965242, −6.38038409846357194633931114598, −5.48609360755794780059129163851, −4.38715563511724473984896445405, −3.91004007574990409902234762171, −2.58423396936077715589664848412, −1.34729633229788680271414708072,
1.29740771872686422092232303469, 2.39572908894048676199247396995, 3.01176110627204262949851672611, 4.67165921662930019052676881143, 5.50992780352555602462229344153, 6.32866234294700567987733171649, 6.83920222662936849447471256004, 7.80873218189820654173280177863, 8.316714233475064938423614709278, 9.069461020759765572372245277468