L(s) = 1 | − 5-s + (1 + i)13-s + (1 − i)17-s + 25-s + 2i·29-s + (1 − i)37-s + 2·41-s + i·49-s + (−1 − i)53-s + (−1 − i)65-s + (−1 − i)73-s + (−1 + i)85-s + 2i·89-s + (−1 + i)97-s − 2i·109-s + ⋯ |
L(s) = 1 | − 5-s + (1 + i)13-s + (1 − i)17-s + 25-s + 2i·29-s + (1 − i)37-s + 2·41-s + i·49-s + (−1 − i)53-s + (−1 − i)65-s + (−1 − i)73-s + (−1 + i)85-s + 2i·89-s + (−1 + i)97-s − 2i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9792370769\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9792370769\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 2iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 - 2T + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - 2iT - T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−9.514494561939992989969555017758, −9.027056338595591550620069265655, −8.075038895783426469924256153369, −7.39065439420921194272443620685, −6.64238716791755925430759070677, −5.59502603952297633276154014009, −4.58575111085215552545235867765, −3.77350033148560231280538830435, −2.86918491009519607751869513752, −1.22250185924146832506956428193,
1.05868578085971676079954382992, 2.79545552444716794051680904709, 3.72863568514383898177399059257, 4.43913993598052819559930810697, 5.72266021236676304890805641597, 6.28416467723877955480286354843, 7.61931739906611022014648553378, 7.970727653561362937937201211123, 8.684468994026878119935643016065, 9.789834312403162101504104368777