L(s) = 1 | + 2.85i·7-s + 1.85·11-s + 0.854i·13-s − 1.14i·17-s − 2·19-s − 4.85i·23-s − 3.70·29-s + 2.70·31-s + 5.85i·37-s − 11.5·41-s + 0.854i·43-s + 6.70i·47-s − 1.14·49-s + 4.85i·53-s − 1.14·59-s + ⋯ |
L(s) = 1 | + 1.07i·7-s + 0.559·11-s + 0.236i·13-s − 0.277i·17-s − 0.458·19-s − 1.01i·23-s − 0.688·29-s + 0.486·31-s + 0.962i·37-s − 1.80·41-s + 0.130i·43-s + 0.978i·47-s − 0.163·49-s + 0.666i·53-s − 0.149·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9003343668\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9003343668\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 2.85iT - 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 - 0.854iT - 13T^{2} \) |
| 17 | \( 1 + 1.14iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 4.85iT - 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 - 5.85iT - 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 0.854iT - 43T^{2} \) |
| 47 | \( 1 - 6.70iT - 47T^{2} \) |
| 53 | \( 1 - 4.85iT - 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 - 0.854T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 2.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 6.70iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.349913735327346582958026400749, −7.39341629588402712157193459307, −6.58171354082493854533482156141, −6.17673729161197891453750300823, −5.31491082978398197103130200962, −4.66939831252681329352168391057, −3.86437782856170392515247448501, −2.91083784088174534269873659026, −2.24077725466411603929953345379, −1.26392896710841313195533349435,
0.21202165684284673663922051451, 1.32576602300885447917966658340, 2.16157800486398701998417842853, 3.50263357319905574674097347266, 3.75230822145319863106847380486, 4.68377260700630546865821227095, 5.41387041403962100526961760154, 6.25379993171981873418718185233, 6.90472079087382840377418103470, 7.47229270850675541311716452211