L(s) = 1 | + 2.23·5-s + (2.23 − 1.41i)7-s + 5.65i·11-s + 4.47·13-s − 3.16i·17-s − 3.16i·19-s − 4·23-s + 5.00·25-s − 2.82i·29-s − 6.32i·31-s + (5.00 − 3.16i)35-s − 9.89i·37-s + 4.47·41-s + 1.41i·43-s − 9.48i·47-s + ⋯ |
L(s) = 1 | + 0.999·5-s + (0.845 − 0.534i)7-s + 1.70i·11-s + 1.24·13-s − 0.766i·17-s − 0.725i·19-s − 0.834·23-s + 1.00·25-s − 0.525i·29-s − 1.13i·31-s + (0.845 − 0.534i)35-s − 1.62i·37-s + 0.698·41-s + 0.215i·43-s − 1.38i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.939426393\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.939426393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
good | 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 3.16iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 9.89iT - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 1.41iT - 43T^{2} \) |
| 47 | \( 1 + 9.48iT - 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 9.48iT - 61T^{2} \) |
| 67 | \( 1 - 7.07iT - 67T^{2} \) |
| 71 | \( 1 - 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 12.6iT - 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.171169444968558517720790744550, −7.33605241244849185732271796514, −6.90764961209249631702776569770, −5.93519193343658286202653019585, −5.31515468900529936014124106379, −4.44419813123629986718182962158, −3.93447790041513994082141400144, −2.44604351512203979385456186122, −1.94968490904991921614020522345, −0.884036760069159266331127140431,
1.17122320042817833095540258239, 1.76094968419382293562871541190, 2.95775605207415749850211660177, 3.63308042877973537761520038359, 4.73505505451342474198198189702, 5.58437022520374303102795641516, 6.07312051065250016848566336575, 6.47143792601795501017632037326, 7.84377011756981697292098795180, 8.444653941015290925014432522569