L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 2i·7-s − i·8-s − 9-s − 2·11-s − i·12-s + 6i·13-s + 2·14-s + 16-s − 4i·17-s − i·18-s + 2·21-s − 2i·22-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.603·11-s − 0.288i·12-s + 1.66i·13-s + 0.534·14-s + 0.250·16-s − 0.970i·17-s − 0.235i·18-s + 0.436·21-s − 0.426i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9226333152\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9226333152\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.585255844645099596686488605772, −7.64014092366496130565654014323, −7.09559804780516375487050331039, −6.39976270911073491713134510731, −5.45048340988503034552991537286, −4.67395803607590011113284780329, −4.14052932573854012375675609409, −3.19815331382713430407515435618, −1.92646857082967767835057903319, −0.30887519187832239194676175518,
1.06158891145118964567898232295, 2.17761565878045051384221333498, 2.92410289600648693419139718745, 3.69479499315545790979772366330, 4.98181983487679394974802984851, 5.56934855277317742692276713836, 6.22987147982571618024391263602, 7.33749370831364866222088395338, 8.164687724289822153202893941031, 8.450611534843835839013649290509