L(s) = 1 | + 5-s + i·9-s − 2i·13-s + 25-s + (1 − i)29-s + 37-s + i·45-s + i·49-s + (−1 + i)53-s + (1 + i)61-s − 2i·65-s + (−1 − i)73-s − 81-s + (−1 + i)89-s + 2·97-s + ⋯ |
L(s) = 1 | + 5-s + i·9-s − 2i·13-s + 25-s + (1 − i)29-s + 37-s + i·45-s + i·49-s + (−1 + i)53-s + (1 + i)61-s − 2i·65-s + (−1 − i)73-s − 81-s + (−1 + i)89-s + 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.520859467\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.520859467\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-1 + i)T - iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (-1 - i)T + iT^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (1 - i)T - iT^{2} \) |
| 97 | \( 1 - 2T + T^{2} \) |
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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
−8.913067598529495488702094163069, −7.995737472845140081692412820653, −7.67040064438679314514119852193, −6.47376079546832560673368628954, −5.78028760957398732186724775059, −5.20662776530580697136081023284, −4.37015151238388350983553211857, −2.96644912324728096945516213202, −2.45970070691212981498848123866, −1.14820562126073964932029727565,
1.31579764839845224557079622792, 2.23163055592869945829565136046, 3.32047058723932312643792262325, 4.29669721301503063957824826984, 5.06694596616062340701323294831, 6.11218130779767829218663894678, 6.61266789467305997149464159372, 7.14994014115242988470709014104, 8.496464401155093086501324871858, 8.981561352064478990295532479198