L(s) = 1 | − i·2-s − 4-s + i·8-s − 9-s + 0.618i·13-s + 16-s + 1.61i·17-s + i·18-s + 0.618·26-s + 1.61·29-s − i·32-s + 1.61·34-s + 36-s + 1.61i·37-s + 0.618·41-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + i·8-s − 9-s + 0.618i·13-s + 16-s + 1.61i·17-s + i·18-s + 0.618·26-s + 1.61·29-s − i·32-s + 1.61·34-s + 36-s + 1.61i·37-s + 0.618·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8599980639\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8599980639\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.618iT - T^{2} \) |
| 17 | \( 1 - 1.61iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.61iT - T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 0.618iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.618iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 1.61T + T^{2} \) |
| 97 | \( 1 - 1.61iT - T^{2} \) |
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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
−9.154101812059798131991724460705, −8.377133069776784566232893356696, −8.065747707974702558488042021195, −6.61735618993872177850138455123, −5.94066651665368945124182990021, −4.97651031618491700260342894964, −4.18288672127242375762140413149, −3.27777613987017360413116688629, −2.43083820172957114867873953669, −1.32037444264783487904565903411,
0.62828097749147127048723355992, 2.60856274578672415848975273332, 3.46840180422705486251243593380, 4.65568841957767370800265542642, 5.28402419773630207685451165134, 6.01201859067437568955349665841, 6.81221823988510534100444383719, 7.59186397913319923042942574990, 8.249922078591038801952009196817, 8.966844480668144748648531848841