| L(s) = 1 | − i·2-s − 4-s + (−2.27 − 1.35i)7-s + i·8-s − 2.71i·11-s − 6.54i·13-s + (−1.35 + 2.27i)14-s + 16-s − 1.53·17-s + 2.30i·19-s − 2.71·22-s − 3.83i·23-s − 6.54·26-s + (2.27 + 1.35i)28-s + 3.83i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.858 − 0.512i)7-s + 0.353i·8-s − 0.817i·11-s − 1.81i·13-s + (−0.362 + 0.607i)14-s + 0.250·16-s − 0.371·17-s + 0.528i·19-s − 0.577·22-s − 0.799i·23-s − 1.28·26-s + (0.429 + 0.256i)28-s + 0.711i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3811740276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3811740276\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.27 + 1.35i)T \) |
| good | 11 | \( 1 + 2.71iT - 11T^{2} \) |
| 13 | \( 1 + 6.54iT - 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 2.30iT - 19T^{2} \) |
| 23 | \( 1 + 3.83iT - 23T^{2} \) |
| 29 | \( 1 - 3.83iT - 29T^{2} \) |
| 31 | \( 1 - 3.25iT - 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 - 0.468T + 43T^{2} \) |
| 47 | \( 1 - 9.11T + 47T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 4.78iT - 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2.30iT - 71T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 - 16.9iT - 97T^{2} \) |
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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
−8.434285314051787939006170934166, −7.49011710639642934662094036458, −6.64343796702088489030052197270, −5.76733963735182311292784449402, −5.11369619577876174481102288874, −3.93051422437978095487167499826, −3.29866918555952496522794985029, −2.61777538240723846530650680493, −1.12061843740778628208891698941, −0.12878817138318499981805586129,
1.73607955160185689410687352916, 2.74184197014620646164422497747, 3.99876219675331723756603013257, 4.50981084447557890230102656544, 5.57047802680550372546300573911, 6.25525608669027424936171126656, 7.03912932225303191655368095366, 7.35800746058932671910307824408, 8.640870165489566632485427516220, 9.075915280690686580465296448079
