| L(s) = 1 | + i·2-s + 0.554·3-s − 4-s + 1.35i·5-s + 0.554i·6-s + i·7-s − i·8-s − 2.69·9-s − 1.35·10-s − 1.80i·11-s − 0.554·12-s − 14-s + 0.753i·15-s + 16-s − 5.29·17-s − 2.69i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.320·3-s − 0.5·4-s + 0.606i·5-s + 0.226i·6-s + 0.377i·7-s − 0.353i·8-s − 0.897·9-s − 0.429·10-s − 0.543i·11-s − 0.160·12-s − 0.267·14-s + 0.194i·15-s + 0.250·16-s − 1.28·17-s − 0.634i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4111044295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4111044295\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.554T + 3T^{2} \) |
| 5 | \( 1 - 1.35iT - 5T^{2} \) |
| 11 | \( 1 + 1.80iT - 11T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 1.95iT - 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 - 5.96T + 29T^{2} \) |
| 31 | \( 1 + 3.63iT - 31T^{2} \) |
| 37 | \( 1 + 7.75iT - 37T^{2} \) |
| 41 | \( 1 - 0.0392iT - 41T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 + 4.07iT - 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 2.22iT - 67T^{2} \) |
| 71 | \( 1 - 7.78iT - 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 2.85iT - 83T^{2} \) |
| 89 | \( 1 - 6.61iT - 89T^{2} \) |
| 97 | \( 1 + 3.64iT - 97T^{2} \) |
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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
−8.652158831357139860109231001642, −8.161593409094654253495212502748, −7.27313600757244275236927910071, −6.36175487954147506287142319142, −5.94933979237075491237758859067, −4.97804873821139567595690437682, −3.94162004090687334023183075306, −3.02631918668772820464595965143, −2.12649896793042672155515590134, −0.13246974357720043123600783441,
1.30593983650819442596433920172, 2.44669685247006817686966996771, 3.22409875790898350669266296096, 4.47199174112617466366401568176, 4.76478951045180182653198993433, 5.98214131614288395823985899661, 6.81845387516363576747238290125, 7.88140294792055127153076635400, 8.593163921572338027277046314961, 9.043619091432452444621945358157
