L(s) = 1 | + 3-s + (−2 − i)5-s + 4i·7-s + 9-s − 4·13-s + (−2 − i)15-s − 8i·19-s + 4i·21-s + 4i·23-s + (3 + 4i)25-s + 27-s + 6i·29-s + 8·31-s + (4 − 8i)35-s + 4·37-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.894 − 0.447i)5-s + 1.51i·7-s + 0.333·9-s − 1.10·13-s + (−0.516 − 0.258i)15-s − 1.83i·19-s + 0.872i·21-s + 0.834i·23-s + (0.600 + 0.800i)25-s + 0.192·27-s + 1.11i·29-s + 1.43·31-s + (0.676 − 1.35i)35-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + (2 + i)T \) |
good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.423268147565705613987388896913, −7.43358189713671177990028362628, −6.93753104841989916125780631912, −5.83705937186136313940050962647, −4.85853494379528576117398194872, −4.61948640755906979949738409316, −3.14433288177430501563170938341, −2.79354729987096860810481363014, −1.62165873283666324814083796754, 0,
1.31757826245297769714860437020, 2.65287662465497880331242984052, 3.41780476748643744018951276791, 4.31693675529829203231383233395, 4.55599345459936692280238235378, 6.07444055454727640683238062281, 6.80975788516821520580556347131, 7.57560801113113330696861616299, 7.87536061749223375012990524674