| L(s) = 1 | − i·2-s − 2i·3-s − 4-s − 2·6-s − i·7-s + i·8-s − 9-s − 11-s + 2i·12-s − 4i·13-s − 14-s + 16-s + i·18-s + 4·19-s − 2·21-s + i·22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 0.301·11-s + 0.577i·12-s − 1.10i·13-s − 0.267·14-s + 0.250·16-s + 0.235i·18-s + 0.917·19-s − 0.436·21-s + 0.213i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.105078163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.105078163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.061146837409206861311940968066, −7.23345929626212748966836536876, −6.76819731821225583985873233538, −5.60161142452796111599114351892, −5.11193981738037520493063668035, −3.88009763141889101053290266785, −3.12487244622707465681327377196, −2.17813940416829836819236984710, −1.27180840645154899122025656630, −0.33711306904813403695903105835,
1.57092755178095351879557476020, 2.97833875110423585315257832570, 3.79382735738498953232004012135, 4.61575154933212041110754030755, 5.13133380071392062450343136350, 5.90568131516190423655155581512, 6.81558203958049030034712086005, 7.42876011783554666035557562147, 8.412190586380507843156572965288, 8.985425999812591775857981830845
