| L(s) = 1 | − i·3-s + 3.79i·5-s − 2.15·7-s − 9-s + 2.54i·11-s + 1.95i·13-s + 3.79·15-s + 0.224·17-s + 0.224i·19-s + 2.15i·21-s + 2.82·23-s − 9.42·25-s + i·27-s + 2.62i·29-s + 1.84·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.69i·5-s − 0.816·7-s − 0.333·9-s + 0.766i·11-s + 0.542i·13-s + 0.980·15-s + 0.0545·17-s + 0.0515i·19-s + 0.471i·21-s + 0.589·23-s − 1.88·25-s + 0.192i·27-s + 0.487i·29-s + 0.330·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6122858857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6122858857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.79iT - 5T^{2} \) |
| 7 | \( 1 + 2.15T + 7T^{2} \) |
| 11 | \( 1 - 2.54iT - 11T^{2} \) |
| 13 | \( 1 - 1.95iT - 13T^{2} \) |
| 17 | \( 1 - 0.224T + 17T^{2} \) |
| 19 | \( 1 - 0.224iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 2.62iT - 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 - 5.18iT - 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 10.6iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 8.46iT - 61T^{2} \) |
| 67 | \( 1 - 14.7iT - 67T^{2} \) |
| 71 | \( 1 - 4.31T + 71T^{2} \) |
| 73 | \( 1 + 5.97T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 1.42T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−9.145264721083196790081614166666, −8.198034407470153905265500038577, −7.12656761927456687275625753131, −7.00904369671613839486598626291, −6.36216679296439169807281242852, −5.49382299441818044735193284561, −4.25254038488250051801490204794, −3.26100083371847024702532556486, −2.69852835696930209457149265005, −1.69456127673105106080537685560,
0.19768940058076976072028292906, 1.22586060837092614438850320328, 2.75556945518502413928690731623, 3.66204078901912113184456364181, 4.46717900213930326518542548624, 5.26551339275002152781421227772, 5.80009143469457618051996917706, 6.69366305270431588945273336419, 7.935675980787709763675874445589, 8.399818968050937943904208783479
