L(s) = 1 | − 1.73·3-s − 3.46i·5-s + 2i·7-s + 2.99·9-s + 3.46·11-s + 5.99i·15-s − 3.46i·21-s − 6.99·25-s − 5.19·27-s − 10.3i·29-s − 10i·31-s − 5.99·33-s + 6.92·35-s − 10.3i·45-s + 3·49-s + ⋯ |
L(s) = 1 | − 1.00·3-s − 1.54i·5-s + 0.755i·7-s + 0.999·9-s + 1.04·11-s + 1.54i·15-s − 0.755i·21-s − 1.39·25-s − 1.00·27-s − 1.92i·29-s − 1.79i·31-s − 1.04·33-s + 1.17·35-s − 1.54i·45-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.726029 - 0.726029i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.726029 - 0.726029i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.870932956974164701728770604964, −9.368769219349192003948541520197, −8.515705662680264033963656432675, −7.57002028151787001773175902478, −6.22281190656349411477149969298, −5.73428704221590813478382901402, −4.70513787943132556190126113640, −4.03924374989051954847480539257, −1.94579469139449183739562668870, −0.65620606335608006943425025629,
1.41647093101350920286495424003, 3.17149221172108981476098577944, 4.06847856522409827648720879197, 5.25895141410010010667904006152, 6.48524605960970495130017039276, 6.82803893839965645554382810325, 7.52835756594407838306636187986, 8.991429082883947291173430204248, 10.06051219561991928830921454848, 10.68926374993871336664540652328