L(s) = 1 | + (0.707 + 1.70i)5-s + (0.707 − 0.707i)9-s − 1.41i·13-s + i·17-s + (−1.70 + 1.70i)25-s + (0.292 + 0.707i)29-s + (−0.707 + 0.292i)37-s + (−0.292 + 0.707i)41-s + (1.70 + 0.707i)45-s + (−0.707 − 0.707i)49-s + (−1 − i)53-s + (0.707 − 1.70i)61-s + (2.41 − 1.00i)65-s + (0.292 + 0.707i)73-s − 1.00i·81-s + ⋯ |
L(s) = 1 | + (0.707 + 1.70i)5-s + (0.707 − 0.707i)9-s − 1.41i·13-s + i·17-s + (−1.70 + 1.70i)25-s + (0.292 + 0.707i)29-s + (−0.707 + 0.292i)37-s + (−0.292 + 0.707i)41-s + (1.70 + 0.707i)45-s + (−0.707 − 0.707i)49-s + (−1 − i)53-s + (0.707 − 1.70i)61-s + (2.41 − 1.00i)65-s + (0.292 + 0.707i)73-s − 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.204028640\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.204028640\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - iT \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
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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
−10.14157946350446174712644323333, −9.735567167185059470290306289593, −8.453606198238965955869943384111, −7.53648746002321060518381097675, −6.66608437287159737844845453324, −6.20652038639220617237305203036, −5.16832660182240855445452260040, −3.62543208897488258804148033244, −3.06603280139541375563812241554, −1.76430190568777734531224439304,
1.34508858697204042613597973255, 2.27551395496187690721884307151, 4.16958569083548456535024817943, 4.75791733109002344681738599399, 5.50061292456226943874737950690, 6.58803543601726379848400787562, 7.56676357134748393937685000414, 8.488504426115647243413030115251, 9.276142860031979840965009840579, 9.658933927688974309156026261663