L(s) = 1 | + 3i·3-s + 1.64·5-s + (15.2 − 10.4i)7-s − 9·9-s − 20.3·11-s + 13.0·13-s + 4.92i·15-s + 23.9i·17-s + 87.7i·19-s + (31.4 + 45.8i)21-s − 73.6i·23-s − 122.·25-s − 27i·27-s − 58.9i·29-s − 124.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.146·5-s + (0.824 − 0.565i)7-s − 0.333·9-s − 0.557·11-s + 0.279·13-s + 0.0847i·15-s + 0.341i·17-s + 1.05i·19-s + (0.326 + 0.476i)21-s − 0.667i·23-s − 0.978·25-s − 0.192i·27-s − 0.377i·29-s − 0.723·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.191930337\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.191930337\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + (-15.2 + 10.4i)T \) |
good | 5 | \( 1 - 1.64T + 125T^{2} \) |
| 11 | \( 1 + 20.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 87.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 73.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 58.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 56.5iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 135. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 259.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 217.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 529. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 685. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 149.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 409.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 885. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 269. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 902. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 623. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 986. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 179. iT - 9.12e5T^{2} \) |
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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
−9.744299856967247408237560769786, −8.617632847328912895650487208393, −8.062015053869370952418919941603, −7.25419583052147151957309178043, −6.06699763984811911443357196944, −5.36449950767307375097497586171, −4.36725573076353895778963669899, −3.71257176500235302799490641581, −2.39977841161562264415483551900, −1.25302355779875459602839660477,
0.26718651316517011502301288663, 1.65014449366000705964978631954, 2.43148025395089575084666267358, 3.60307989215747981193573186984, 4.96943691020573271923431324580, 5.47274991593613362052243159441, 6.49284124536724728816914726599, 7.40532872622863869352245371085, 8.054699241725022549100793129213, 8.862497796632112375039941587638