| L(s) = 1 | + (−2 − i)5-s − i·7-s − 4i·13-s − 2i·17-s − 8·19-s + 8i·23-s + (3 + 4i)25-s − 8·29-s − 4·31-s + (−1 + 2i)35-s − 8i·37-s + 12·41-s + 8i·43-s + 4i·47-s − 49-s + ⋯ |
| L(s) = 1 | + (−0.894 − 0.447i)5-s − 0.377i·7-s − 1.10i·13-s − 0.485i·17-s − 1.83·19-s + 1.66i·23-s + (0.600 + 0.800i)25-s − 1.48·29-s − 0.718·31-s + (−0.169 + 0.338i)35-s − 1.31i·37-s + 1.87·41-s + 1.21i·43-s + 0.583i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6913674992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6913674992\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 4T + 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.202114794783466616145419020550, −7.58572814827864194059476194669, −7.27811960694866264961540900248, −6.06075311253779056868295015896, −5.49313238358292633904719100518, −4.55022549749723506801316460637, −3.90725478816503839315019275517, −3.20760516573395121618763039713, −2.03355944636581295610418996941, −0.805567807020697005653205270531,
0.24301156939256425401014478676, 1.92941188939022906466980902041, 2.58003541959055927515434126948, 3.86216482990394631660876407120, 4.14702199047817695435539275451, 5.07765990034819504723366533938, 6.21355526195421843031011328976, 6.60847851729709563504326879697, 7.35761897900818770359524397274, 8.204950846556536979289426268798
