| L(s) = 1 | + i·2-s − 4-s − 3.41·5-s + (0.143 − 2.64i)7-s − i·8-s − 3.41i·10-s + 1.10i·11-s + 6.37i·13-s + (2.64 + 0.143i)14-s + 16-s − 7.75·17-s + i·19-s + 3.41·20-s − 1.10·22-s − 7.32i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.52·5-s + (0.0541 − 0.998i)7-s − 0.353i·8-s − 1.07i·10-s + 0.334i·11-s + 1.76i·13-s + (0.706 + 0.0383i)14-s + 0.250·16-s − 1.88·17-s + 0.229i·19-s + 0.762·20-s − 0.236·22-s − 1.52i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8406146947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8406146947\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-0.143 + 2.64i)T \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 - 1.10iT - 11T^{2} \) |
| 13 | \( 1 - 6.37iT - 13T^{2} \) |
| 17 | \( 1 + 7.75T + 17T^{2} \) |
| 23 | \( 1 + 7.32iT - 23T^{2} \) |
| 29 | \( 1 + 6.66iT - 29T^{2} \) |
| 31 | \( 1 - 1.05iT - 31T^{2} \) |
| 37 | \( 1 - 7.54T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 + 2.20T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 - 12.9iT - 53T^{2} \) |
| 59 | \( 1 - 7.90T + 59T^{2} \) |
| 61 | \( 1 + 0.763iT - 61T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 + 1.42iT - 71T^{2} \) |
| 73 | \( 1 - 3.02iT - 73T^{2} \) |
| 79 | \( 1 - 6.38T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 8.74T + 89T^{2} \) |
| 97 | \( 1 + 7.54iT - 97T^{2} \) |
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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
−8.760817310381678964060367889504, −8.217403561150382572250361282263, −7.35978557676059509536925223413, −6.85256433359822370692301225878, −6.31846340137730543819378379779, −4.54817223126240601612029794923, −4.44562858502331502215741574567, −3.80092663359287693757886813017, −2.24389204526117953547543717162, −0.53432640786725455051489887684,
0.60634701788439146227432717749, 2.22542558734215483293007035439, 3.20956369422264877351403973465, 3.76820595463071383447627980038, 4.90087741362518882648368611168, 5.46786388972886829555899472019, 6.64481464184327012483880839821, 7.61892504361790649546794322401, 8.297376415403373563298897928140, 8.762288400753644006461306863438
