L(s) = 1 | + i·2-s + (−1.06 + 1.36i)3-s − 4-s + (−1.36 − 1.06i)6-s + (2.62 − 0.294i)7-s − i·8-s + (−0.716 − 2.91i)9-s − 1.43i·11-s + (1.06 − 1.36i)12-s + 4.73i·13-s + (0.294 + 2.62i)14-s + 16-s + 2.59·17-s + (2.91 − 0.716i)18-s + 2.72i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.616 + 0.786i)3-s − 0.5·4-s + (−0.556 − 0.436i)6-s + (0.993 − 0.111i)7-s − 0.353i·8-s + (−0.238 − 0.971i)9-s − 0.431i·11-s + (0.308 − 0.393i)12-s + 1.31i·13-s + (0.0786 + 0.702i)14-s + 0.250·16-s + 0.630·17-s + (0.686 − 0.168i)18-s + 0.625i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.197179898\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.197179898\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.06 - 1.36i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.62 + 0.294i)T \) |
good | 11 | \( 1 + 1.43iT - 11T^{2} \) |
| 13 | \( 1 - 4.73iT - 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 19 | \( 1 - 2.72iT - 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 5.91iT - 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 + 0.432T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.69iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.05iT - 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 + 13.2iT - 71T^{2} \) |
| 73 | \( 1 - 7.58iT - 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 3.90T + 89T^{2} \) |
| 97 | \( 1 - 14.0iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.29136606248714454784362885674, −9.271216747618421753460424183853, −8.707917004420322322904494173300, −7.72130142409180694867605140403, −6.80961414585876860261903693050, −5.89134232057265525008172428511, −5.11825940542412547908100144445, −4.36760561986899344439775804188, −3.43265540813750440213394543886, −1.41374266327413573039296007045,
0.64688343361717860049380479484, 1.86885243236405861684298872184, 2.88436279169858277173064115710, 4.40556433566886486484527180350, 5.23179711281064534593059300231, 5.96207537463017239872487176178, 7.23411625434205299273878760606, 7.939284493433183976975446177813, 8.586972301811642052135858007300, 9.845770861484367106862379949468