| L(s) = 1 | + i·4-s − i·5-s − 7-s − 16-s + (1 − i)17-s + 20-s + 23-s − i·28-s + 29-s + (1 − i)31-s + i·35-s + (1 + i)37-s + (1 + i)41-s + (−1 − i)47-s + 53-s + ⋯ |
| L(s) = 1 | + i·4-s − i·5-s − 7-s − 16-s + (1 − i)17-s + 20-s + 23-s − i·28-s + 29-s + (1 − i)31-s + i·35-s + (1 + i)37-s + (1 + i)41-s + (−1 − i)47-s + 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.093570875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093570875\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - iT^{2} \) |
| 5 | \( 1 + iT - T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + (-1 + i)T - iT^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (1 + i)T + iT^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 + iT - T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.305134393330529707605753632343, −8.287400472825736004553177689436, −7.79501360615056756381825861626, −6.86154607734526134628713924083, −6.14832004139168365807020156519, −4.97319908251020754179885928260, −4.42256626606445576128210246825, −3.24298088113248238688516250838, −2.72817665679783159836025001549, −0.930357227656702054880520072378,
1.15555129381057699301323505858, 2.59849964297284911130658777640, 3.27368605251826183374782963944, 4.39300616518608718716527575519, 5.45853757730507149171352741613, 6.23803972827246826326582240184, 6.65544577940571423129189195672, 7.46470510273772524571081124296, 8.522778225583319620606362124773, 9.439010897641020007044047736536
