| L(s) = 1 | − 0.873·3-s + 0.992·7-s − 2.23·9-s + 1.83·11-s − 3.28·13-s + 6.63·17-s − 5.64·19-s − 0.866·21-s + 23-s + 4.57·27-s + 2.01·29-s − 0.315·31-s − 1.60·33-s + 3.07·37-s + 2.87·39-s − 1.34·41-s + 5.97·43-s − 0.306·47-s − 6.01·49-s − 5.79·51-s + 6.98·53-s + 4.92·57-s + 9.49·59-s + 5.56·61-s − 2.22·63-s − 0.853·67-s − 0.873·69-s + ⋯ |
| L(s) = 1 | − 0.504·3-s + 0.375·7-s − 0.745·9-s + 0.552·11-s − 0.911·13-s + 1.60·17-s − 1.29·19-s − 0.189·21-s + 0.208·23-s + 0.880·27-s + 0.374·29-s − 0.0565·31-s − 0.278·33-s + 0.505·37-s + 0.459·39-s − 0.210·41-s + 0.911·43-s − 0.0446·47-s − 0.859·49-s − 0.811·51-s + 0.959·53-s + 0.652·57-s + 1.23·59-s + 0.712·61-s − 0.279·63-s − 0.104·67-s − 0.105·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.356493842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.356493842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.873T + 3T^{2} \) |
| 7 | \( 1 - 0.992T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 - 6.63T + 17T^{2} \) |
| 19 | \( 1 + 5.64T + 19T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 + 0.315T + 31T^{2} \) |
| 37 | \( 1 - 3.07T + 37T^{2} \) |
| 41 | \( 1 + 1.34T + 41T^{2} \) |
| 43 | \( 1 - 5.97T + 43T^{2} \) |
| 47 | \( 1 + 0.306T + 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 - 9.49T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 + 0.853T + 67T^{2} \) |
| 71 | \( 1 - 0.797T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 3.62T + 79T^{2} \) |
| 83 | \( 1 - 17.1T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−8.937444013315679090196714360730, −8.251811008865871152953630552493, −7.48220458013780818473843956123, −6.59623889210469575927320603456, −5.82151220798693941065202989916, −5.13491409309947905311544301728, −4.27989651392175578680020231319, −3.19616597593089960740570363649, −2.15908530356660004645417972603, −0.76984366552197518424516199839,
0.76984366552197518424516199839, 2.15908530356660004645417972603, 3.19616597593089960740570363649, 4.27989651392175578680020231319, 5.13491409309947905311544301728, 5.82151220798693941065202989916, 6.59623889210469575927320603456, 7.48220458013780818473843956123, 8.251811008865871152953630552493, 8.937444013315679090196714360730
