| L(s) = 1 | + 2i·2-s − 4·4-s − 8i·8-s − 10i·13-s + 16·16-s + 16i·17-s + 20·26-s + 40·29-s + 32i·32-s − 32·34-s + 70i·37-s + 80·41-s − 49·49-s + 40i·52-s + 56i·53-s + ⋯ |
| L(s) = 1 | + i·2-s − 4-s − i·8-s − 0.769i·13-s + 16-s + 0.941i·17-s + 0.769·26-s + 1.37·29-s + i·32-s − 0.941·34-s + 1.89i·37-s + 1.95·41-s − 0.999·49-s + 0.769i·52-s + 1.05i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.454938921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.454938921\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 10iT - 169T^{2} \) |
| 17 | \( 1 - 16iT - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 529T^{2} \) |
| 29 | \( 1 - 40T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 70iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 80T + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.20e3T^{2} \) |
| 53 | \( 1 - 56iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 22T + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 110iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 160T + 7.92e3T^{2} \) |
| 97 | \( 1 - 130iT - 9.40e3T^{2} \) |
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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
−10.07982587195357525504705340380, −9.182783159808963125587651340582, −8.258886436307642805109100082212, −7.78794276866251576123792979772, −6.64465359542742544264425326525, −6.01134698761195672394974302811, −5.02596265131659000602478233837, −4.12815353150842717986766337941, −2.95007254879951885496594675353, −1.05301674033675730715462372274,
0.59508477684207740752824866212, 2.00085995923988687091411433567, 3.01718158822779872840855952951, 4.15863202857464013306969483543, 4.93106542204228106335146193714, 6.02517159416098460207701642273, 7.19176073931653637401584545613, 8.157639555521874290071132518076, 9.136376250783893696167736405410, 9.586475660256515602254086037809
