| L(s) = 1 | − i·3-s + 5-s + 8·9-s + 17i·11-s + 24·13-s − i·15-s + 17-s + 7i·19-s − 7i·23-s − 24·25-s − 17i·27-s + 24·29-s + 41i·31-s + 17·33-s − 49·37-s + ⋯ |
| L(s) = 1 | − 0.333i·3-s + 0.200·5-s + 0.888·9-s + 1.54i·11-s + 1.84·13-s − 0.0666i·15-s + 0.0588·17-s + 0.368i·19-s − 0.304i·23-s − 0.959·25-s − 0.629i·27-s + 0.827·29-s + 1.32i·31-s + 0.515·33-s − 1.32·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.328205999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328205999\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + iT - 9T^{2} \) |
| 5 | \( 1 - T + 25T^{2} \) |
| 11 | \( 1 - 17iT - 121T^{2} \) |
| 13 | \( 1 - 24T + 169T^{2} \) |
| 17 | \( 1 - T + 289T^{2} \) |
| 19 | \( 1 - 7iT - 361T^{2} \) |
| 23 | \( 1 + 7iT - 529T^{2} \) |
| 29 | \( 1 - 24T + 841T^{2} \) |
| 31 | \( 1 - 41iT - 961T^{2} \) |
| 37 | \( 1 + 49T + 1.36e3T^{2} \) |
| 41 | \( 1 + 48T + 1.68e3T^{2} \) |
| 43 | \( 1 - 24iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 55iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 25T + 2.80e3T^{2} \) |
| 59 | \( 1 + 17iT - 3.48e3T^{2} \) |
| 61 | \( 1 + T + 3.72e3T^{2} \) |
| 67 | \( 1 + 65iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 96iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 95T + 5.32e3T^{2} \) |
| 79 | \( 1 - 41iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 72iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 95T + 7.92e3T^{2} \) |
| 97 | \( 1 - 144T + 9.40e3T^{2} \) |
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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
−9.406719406852110997081303780287, −8.468752889184557884717042750151, −7.75551583448930248518713441600, −6.79243482474875009519434986261, −6.36812905603996545269053181644, −5.16670520761857642507571168290, −4.29688048091104946226541468456, −3.41876164858198585331449951423, −1.93309475750444386976493939301, −1.27849246210382294034575121558,
0.69621390256156682547718037803, 1.82915816527752535301155794634, 3.41245638391770114190056805296, 3.81288014354592734396648854040, 5.04606263594184858383930194039, 5.96497737865809028549843827158, 6.51857873185918851434755613298, 7.64319149725720321384762697754, 8.540410323146462936999189577183, 8.988989986484869830239968770130
