L(s) = 1 | − i·3-s + i·7-s − 9-s − 11-s − 2i·13-s − 6·19-s + 21-s + i·23-s + i·27-s − 29-s − 2·31-s + i·33-s + 7i·37-s − 2·39-s − 8·41-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 0.301·11-s − 0.554i·13-s − 1.37·19-s + 0.218·21-s + 0.208i·23-s + 0.192i·27-s − 0.185·29-s − 0.359·31-s + 0.174i·33-s + 1.15i·37-s − 0.320·39-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 3iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.437607445892502240754332991967, −8.054314537523750635667214948919, −7.03677758921550525489848994865, −6.36131518541810305036280857163, −5.54235321092172402621071320641, −4.72152396439568378696730208417, −3.53913857488412270655402271833, −2.57639747745573851900917073996, −1.59015895549160105601758196548, 0,
1.77147715172929459234084824411, 2.90236480782709475829090676440, 4.00026074984720063504882424907, 4.54705737883419390992021308708, 5.55620179496014680898794398032, 6.38985305186693732324144016934, 7.21360091994638388362535412982, 8.083623376991630112615036332161, 8.888228717636609748923984109824