L(s) = 1 | + 4i·3-s − 5i·5-s − 4·7-s − 7·9-s + 20·15-s − 16i·21-s + 44·23-s − 25·25-s + 8i·27-s + 22i·29-s + 20i·35-s − 62·41-s + 76i·43-s + 35i·45-s + 4·47-s + ⋯ |
L(s) = 1 | + 1.33i·3-s − i·5-s − 0.571·7-s − 0.777·9-s + 1.33·15-s − 0.761i·21-s + 1.91·23-s − 25-s + 0.296i·27-s + 0.758i·29-s + 0.571i·35-s − 1.51·41-s + 1.76i·43-s + 0.777i·45-s + 0.0851·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.218667633\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.218667633\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 5iT \) |
good | 3 | \( 1 - 4iT - 9T^{2} \) |
| 7 | \( 1 + 4T + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 44T + 529T^{2} \) |
| 29 | \( 1 - 22iT - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 62T + 1.68e3T^{2} \) |
| 43 | \( 1 - 76iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 58iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 116iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 76iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 142T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.709522022090276329098563625636, −9.068973803963312490243754942738, −8.564704776049152732813003457580, −7.38110465145316162593747304437, −6.33636104148734428505017865299, −5.18702157200992367309547424656, −4.80844599379100670912543286831, −3.80199478947581832955197409032, −2.96178156975455524157690426332, −1.21527649783633792059533219864,
0.37852056963888892286509842834, 1.77674798588697002448546208097, 2.76581517558307102902251633139, 3.61863463185476116642467749967, 5.11922337529695829585761535868, 6.24863816301040650669559869645, 6.75858471785717614873811513257, 7.32895438964099767780841807848, 8.129430215047166191015492150163, 9.123698712108264881233629007433